Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 45, pp. 1–38.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.45

EXISTENCE OF UNDERCOMPRESSIVE TRAVELLING WAVES OF A

NON-LOCAL GENERALISED KORTEWEG-DE VRIES-BURGERS EQUATION

FRANZ ACHLEITNER, CARLOTA M. CUESTA, XUBAN DIEZ-IZAGIRRE

Abstract. We study travelling wave solutions of a generalised Korteweg-de Vries-Burgers equa-
tion with a non-local diffusion term and a concave-convex flux. This model equation arises in

the analysis of a shallow water flow by performing formal asymptotic expansions associated to

the triple-deck regularisation (which is an extension of classical boundary layer theory). The re-
sulting non-local operator is a fractional type derivative with order between 1 and 2. Travelling

wave solutions are typically analysed in relation to shock formation in the full shallow water

problem. We show rigorously the existence of travelling waves that, formally, in the limit of
vanishing diffusion and dispersion would give rise to non-classical shocks, that is, shocks that vi-

olate the Lax entropy condition. The proof is based on arguments that are typical in dynamical

systems. The nature of the non-local operator makes this possible, since the resulting travelling
wave equation can be seen as a delayed integro-differential equation. Thus, linearization around

critical points, continuity with respect to parameters and a shooting argument, are the main
steps that we have proved and adapted for solving this problem.

Contents

1. Introduction 2
2. Preliminary results 5
2.1. The non-local operator and some elementary lemmas 5
2.2. Existence of trajectories that satisfy (1.6) and the derivation of (1.14) 7
3. Existence of undercompressive waves: Proof of Theorem 1.1 13
3.1. The set Σu 13
3.2. The sets Σc and Σ+: end of the proof 16
4. Control of the modified problem for small τ : Proof of Theorem 3.5 20
5. Numerical computations 27
6. Appendix: Continuous dependence on τ 29
7. Appendix: Characteristic equations of the linearised problems 31
8. Appendix: The linearized equation 31
8.1. Monotonicity of v for small values of τ 33
Acknowledgments 36
References 37

2020 Mathematics Subject Classification. 47J35, 26A33, 35C07.
Key words and phrases. Non-local evolution equation; fractional derivative; travelling waves;

non-classical shocks.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted December 4, 2024. Published April 30, 2025.

1



2 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

1. Introduction

In this article we study the existence of undercompressive travelling waves for the one-dimen-
sional evolution equation

∂tu+ ∂xu
3 = ∂xDα[u] + τ∂3xu , x ∈ R , t ≥ 0 (1.1)

with τ > 0 and Dα[·] is the non-local operator here applied to a real valued function g,

Dα[g](x) = dα

∫ x

−∞

g′(y)

(x− y)α
dy , with 0 < α < 1 , dα =

1

Γ(1− α)
> 0 . (1.2)

Observe that this operator only acts on the variable x. Here Γ denotes the Gamma function.
Equation (1.1) with α = 1/3 and a flux function (u3 in (1.1)) which is either a quadratic

or a cubic polynomial, has been derived from one (quadratic flux) and two (cubic flux) layer
shallow water flows by performing formal asymptotic expansions in the framework of the triple-
deck boundary layer theory (see [24, 23, 31]). In [31] numerical simulations suggest the existence
of travelling waves that violate the entropy condition in the two-layer model. Such solutions
resemble the inner structure, in such a particular limit, of small amplitude shock waves for the
original shallow water problem, that are not entropic or non-classical (see e.g. [17]). In this
manuscript we aim to study rigorously the existence of such travelling wave solutions for (1.1),
which has a cubic flux and thus corresponds to the two-layer model when α = 1/3.

We recall, that existence and stability of travelling waves for (1.1) with a quadratic flux function
have been established in [2, 4] (τ = 0) and in [1] (τ > 0), see also [11]. The results of [2] and [1]
form part of building blocks in the proof of our main theorem, as we shall describe below.

We recall that hyperbolic conservation laws exhibit discontinuous solutions, whose disconti-
nuities or shocks may travel with constant speed. These solutions belong to the class of weak
solutions. An extra condition is necessary to select a unique solution for the Cauchy problem.
The most common way to derive uniqueness conditions is to use vanishing diffusion arguments
(see e.g. [30]). In particular, for scalar conservation laws admissible shocks result from construct-
ing travelling wave solutions of the regularised parabolic equation. There are, however, other
types of natural regularisations of hyperbolic conservation laws. Moreover, different regularisa-
tions of hyperbolic conservation laws might lead to different weak solutions. Examples describing
this situation are scalar conservation laws with a non-genuinely nonlinear flux (neither convex nor
concave), such as a cubic flux. In this case, shocks violating the classical Lax entropy condition,
see [21], can be constructed by introducing a diffusive-dispersive regularisation. This kind of so-
lutions is defined by Hayes and LeFloch [17] as non-classical shock waves, others refer to them
as undercompressive, see [14] and the references therein. Our aim is thus to show the existence
of such solutions for the non-local version of these regularisations (1.1). For a complementary
study of non-classical shock waves for a scalar conservation law with local diffusion and non-local
dispersion see [27].

We notice that the parameter τ results from a choice in the rescaling. Analogous to [21] we can
consider the equation in the following form

∂tu+ ∂xu
3 = ε∂xDα[u] + δ∂3xu , x ∈ R , t ≥ 0, (1.3)

where ε and δ are positive constants that act as control parameters. This means that depending
on their relative size either diffusion (δ ≪ ε2/α) or dispersion (δ ≫ ε2/α) dominates in the limit
of both ε and δ → 0. The parameter τ results from the scaling (x, t) → (ε1/αx, ε1/αt) so that
τ = δ/ε2/α. It is when this parameter is of order one when we expect to get solutions that violate
the entropy condition.

We introduce the travelling wave variable ξ = x− ct with wave speed c and look for solutions
u(x, t) = ϕ(ξ) of (1.1) which connect two different far-field real values ϕ− and ϕ+. A straightfor-
ward calculation shows that if ϕ depends on x and t only through the travelling wave variable,
then so does Dα[ϕ], and the travelling wave problem becomes

−cϕ′ + (ϕ3)′ = (Dα[ϕ])′ + τϕ′′′. (1.4)



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 3

Here ′ denotes differentiation with respect to ξ. We can then integrate (1.4) with respect to ξ to
arrive at the following travelling wave equation

τϕ′′ +Dα[ϕ] = h(ϕ) , where h(ϕ) := −c(ϕ− ϕ−) + ϕ3 − ϕ3− , (1.5)

where we have used

lim
ξ→−∞

ϕ(ξ) = ϕ− (1.6)

lim
ξ→∞

ϕ(ξ) = ϕ+ . (1.7)

If ϕ′ decays to zero fast enough as ξ → ±∞, then we obtain the Rankine-Hugoniot condition

c =
ϕ3+ − ϕ3−
ϕ+ − ϕ−

= ϕ2+ + ϕ2− + ϕ−ϕ+ , (1.8)

that we assume throughout this manuscript.
One expects that travelling wave solutions correspond to classical shock waves in the limit of the

diffusive and dispersive terms tending to zero (in the right order -dispersion followed by diffusion
tending to zero- or at the right asymptotic rate) if the Lax entropy condition [25, Chapter II.1] is
satisfied, which for the current non-linear flux reads:

3ϕ2+ < c < 3ϕ2− . (1.9)

In this manuscript, however, we investigate the existence of travelling wave solutions that do not
satisfy (1.9). In particular, we shall look for solutions that satisfy

c < 3min{ϕ2−, ϕ2+} . (1.10)

We assume without loss of generality that ϕ+ < ϕ− (observe that the equation is invariant
under the change ϕ→ −ϕ).

The roots of h(ϕ) are ϕ+, ϕ− and ϕc := −(ϕ− + ϕ+). We require that

ϕ+ < ϕc < ϕ− (1.11)

so that

h′(ϕ−) > 0 , h′(ϕc) < 0 and h′(ϕ+) > 0 , (1.12)

which is equivalent to the condition (1.10). We observe that this in particular implies that ϕ− > 0
and ϕ+ < 0.

Under these assumptions, travelling wave solutions of (1.5) with (1.6) and (1.7) correspond to
non-classical shocks in the sense described earlier. On the other hand, solutions that satisfy (1.5)
with (1.6) and

lim
ξ→∞

ϕ(ξ) = ϕc (1.13)

correspond to classical shocks (with the same wave speed). We expect, as in the local case α = 1
(see e.g. [21]), the possibility of solutions satisfying (1.7) to be a distinguished limit in the sense
that there is a unique value of τ that allows such connection, whereas there is an open set of values
of τ that allows solutions to satisfy (1.13). We recall that this last possibility corresponds to the
classical shock admissibility condition (1.9) if we replace ϕ+ by ϕc in the notation. Moreover, if
the only zeros of h where ϕ− and ϕc with (1.12), this would be the only possible travelling wave
solutions that can be constructed in both the local and the non-local case (see [1]).

There is a further necessary condition for existence of (1.5)-(1.7) on the values ϕ− and ϕ+,
namely that ϕ+ + ϕ− > 0, which implies that c > 0 (see (1.8)), in particular. We shall show
this below (see Lemma 2.5), and this is a consequence of the results of [1]. Let us state our main
theorem.

Theorem 1.1 (Existence of undercompressive travelling waves). Let ϕ− and ϕ+ ∈ R such that
(1.11) with ϕc = −(ϕ+ + ϕ−) holds and such that

ϕ+ + ϕ− > 0. (1.14)

Then there exists τ > 0 such that (1.5)-(1.7) has a unique solution (up to a shift in ξ) in C3
b (R).



4 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

We notice that the conditions (1.11) with ϕc = −(ϕ+ + ϕ−) and (1.14) coincide with the
necessary conditions given in [27] for solutions of the Riemann problem to admit a shock wave
that is not entropic in the Lax sense, but that is entropic for the quadratic entropy pair (see
also [25] and references therein, and [21] and [3]). This means that in the limit of zero diffusion
and dispersion (δ → 0+, ε → 0+ with τ = δ/ε2/α of order one in (1.3)) one expects to obtain
undercompressive shocks. We also notice, as we shall discuss in Section 2.2, that this model
does not support solitary waves, because a solution that connects ϕ− to itself ends up to be just
the constant solution ϕ ≡ ϕ−, this is due to the presence of the non-local dissipation term (see
Remark 2.6 in Section 2.2 below).

The uniqueness of τ is an open problem, as well as the dynamic stability of undercompressive
travelling waves. These questions are left for future work. A detailed discussion on the evolution
problem for initial conditions with arbitrary far-field values is out of the scope of this paper. It
is interesting, however, to relate the different ranges of far-field values to a Riemann problem for
the conservation law (i.e. the zero diffusion and zero dispersion case) and whether the expected
solutions are recovered in the limit where δ, ε → 0 while δ/ε2/α is kept of order one in (1.3). A
first step in this direction is to perform numerical simulations that illustrate the Cauchy problem
with an initial condition that connects general constant values. In this regard structure-preserving
methods might be appropriate (see [20] and its applications, see e.g. [19, 18] and references therein)
and this will be part of a future study.

This article is organised as follows. In Section 2, we give some preliminary results. First, in
Section 2.1, we give some lemmas with estimates related to the non-local operator. In Section 2.2,
we establish the existence of solutions of (1.5) that satisfy (1.6) and prove that there are only
three possible behaviours that such trajectories will have as ξ increases: they tend to ϕc or to ϕ+
as ξ → ∞, or they tend to −∞ as ξ → (ξ∗)− for some finite value ξ∗. This allows us to approach
the problem via a shooting argument, with shooting parameter τ . With this in mind, in Section 3
we define three disjoint sets of τ > 0: Σc (the profiles tend to ϕc, thus for such τ ’s the waves are
classical), Σ+ (the profiles tend to ϕ+ and the waves are non-classical) and Σu (the profiles tend
to −∞, and in fact ‘blow up’ at a finite value of ξ). Then we prove Theorem 1.1 by first showing
that Σu is non-empty and open and that Σc is non-empty. Then, we argue by contradiction
assuming that Σ+ is empty, which in particular implies that Σc must be closed. We arrive to a
contradiction by applying continuity with respect to τ , and hence Σ+ must be non-empty, which
proves the behaviour of the desired solutions and their existence, and thus our main theorem.

We note that the proof of Σc being non-empty is very involved: the idea is to use that for
the quadratic case (see [1]) where only classical solutions exist, it was conjectured that for small
enough values of τ the travelling waves are monotone decreasing. This is still a conjecture, but if
this was true, we can look at the problem for a modification of our non-linearity h such that it
is C2 smooth, but that it has only the two zeros ϕ− and ϕc and coincides with h in an interval
of ϕ containing these values. This means that for τ small enough such that the solutions are
monotone, both problems coincide (recall (1.11)), and hence solutions of the original problem for
such small values of τ must tend to ϕc as ξ → ∞, and therefore Σc is non-empty. We do not
prove monotonicity for the modified problem, but it is enough to prove that for small values of
τ the profile solutions remain in the interval where both non-linearities coincide, we show this in
Section 4. Additionally, in this section, for the sake of completeness, we show that if the profiles of
the modified problem are monotone decreasing (and, by similar arguments, for the problem with
a quadratic non-linearity), they decay to ϕc like ξ−α as ξ → ∞, as for the case τ = 0 (see [13]).

In Section 5, we give numerical evidence to illustrate the role of τ in the behaviour of solutions
of (1.5) satisfying (1.6). Indeed, we adapt the value of τ recursively between what appears to be
the two generic behaviours ϕ→ ϕc and ϕ→ −∞, to capture (as long as machine accuracy allows)
the distinguished behaviour ϕ→ ϕ+. We do this for two values of α for illustrative purposes.

In Appendix 6, we prove continuous dependence with respect to τ on finite intervals of existence.
In fact, we only have to formulate our equation as a functional differential equation and apply the
results of [16].



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 5

Finally, we point out that for the proof we need some results on the linearised equations around
ϕ− (some of them are already given in [1], see also [11]) and most importantly around ϕc. In Ap-
pendix 7.1 we give pertinent information about the roots of the characteristic equations associated
to the linearised problems around ϕ− and ϕc.

In particular, we will use repeatedly the following formulation for solutions that decay to ϕc,
or that are close to them in some sense,

τϕ′′ +Dα
ξ0 [ϕ]− h′(ϕc)ϕ = h(ϕ)− h′(ϕc)ϕ− dα

∫ ξ0

−∞

ϕ′(y)

(ξ − y)α
dy ,

where we have split the non-local term by integrating separately on (−∞, ξ0) and on (ξ0, ξ) for
some value ξ0 < ξ. In particular, we use the notation,

Dα
ξ0 [ϕ] := dα

∫ ξ

ξ0

ϕ′(y)

(ξ − y)α
dy ,

which is a Caputo derivative of order α ∈ (0, 1) (see [9, 15]).
In this way and thanks to (1.12), or h′(ϕc) < 0, we can treat the right-hand side as given and

apply a variation of constants type of formulation (see e.g. [9] and [15]) to obtain the solution
implicitly. The study of the corresponding linear equations (the inhomogeneous and the homoge-
neous one) is done in the Appendix 8. In particular, we analyse the fundamental solutions of the
homogeneous equation: we give its behaviour near the initial conditions and in the far-field, and
study its behaviour in several ranges of its domain as τ → 0+ (concluding monotonicity for small
values of τ , in particular). These estimates are crucial to show that Σc is non-empty.

2. Preliminary results

2.1. The non-local operator and some elementary lemmas. Let us first recall some basic
properties of the fractional differential operator Dα[·]. Since it can be written as the convolution
of g′ with θ(x)x−α/Γ(1 − α) (where θ is the Heaviside function), Dα[·] is a pseudo-differential
operator with symbol

ik
√
2π

Γ(1− α)
F
(θ(x)
xα

)
(k) = (bα + iaα sgn(k)) |k|α , (2.1)

i.e. F(Dα[u])(k) = (bα + iaα sgn(k)) |k|αû(k) where F denotes the Fourier transform

F(φ)(k) = φ̂(k) =
1√
2π

∫
R
e−ikxφ(x) dx ,

and the coefficients aα and bα are given by

aα = sin
(απ

2

)
> 0 , bα = cos

(απ
2

)
> 0 , for 0 < α < 1,

(we refer to [6] for the details of the computation to obtain (2.1)). The operator on the right-hand
side of (1.1) is then a pseudo-differential operator with symbol

F(∂xDα) = − (aα − ibα sgn(k)) |k|α+1 , (2.2)

which is dissipative in the sense that the real part of (2.2) is negative.
For s ≥ 0 we shall adopt the following notation for the Sobolev space of square integrable

functions,

Hs := {u ∈ L2(R) : ∥u∥Hs <∞} , ∥u∥Hs := ∥(1 + |k|2)s/2û∥L2(R) ,

and the corresponding homogeneous norm

∥u∥Ḣs := ∥|k|sû∥L2(R) .

Using that (a2α+b
2
α) = 1 it is easy to see that ∥Dαu∥Ḣs = ∥u∥Ḣs+α , and this suggests that one can

interpret Dα[·] as a derivative of order α. We also observe that Dα[·] is a bounded linear operator
from Hs to Hs−α, for all s ≥ 1.

For m ∈ N≥0, let Cm
b (R) denote the set of functions, whose derivatives up to order m are

continuous and bounded. Then one can also infer that Dα[·] is a bounded linear operator from



6 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

C1
b (R) to Cb(R). As explained in [2], this can be easily seen by splitting the domain of integration

in (1.2) into (−∞, x−M ] and [x−M,x] for some positive M > 0. Then integration by parts in
the first integral shows the boundedness of Dα[·]. Moreover, we will need the following improved
estimate.

Lemma 2.1. For α ∈ (0, 1), let x ∈ R and g ∈ C1
b (−∞, x), then for every z ∈ R with z ≤ x,

|Dα[g](z)| ≤ Cα

(
sup

y∈(−∞,z]

|g(y)|
)1−α(

sup
y∈(−∞,z]

|g′(y)|
)α

where

Cα = dα

(2(2α)−α

1− α

)
.

In particular, if g ∈ C1
b (R), then Dα[g] ∈ Cb(R) with

∥Dα[g]∥∞ ≤ Cα∥g′∥α∞∥g∥1−α
∞ .

Proof. The proof is similar to the estimate for Riesz-Feller operators, see e.g. [5, Proposition 2.4].
Let z ≤ x, and let us denote, for simplicity,

A =
(

sup
y∈(−∞,z]

|g(y)|
)

and A′ =
(

sup
y∈(−∞,z]

|g′(y)|
)
.

Then

|Dα[g](z)| = dα

∣∣∣ ∫ ∞

0

g′(z − s)

sα
ds
∣∣∣ ≤ dα

∣∣∣ ∫ M

0

g′(z − s)

sα
ds
∣∣∣+ dα

∣∣∣ ∫ ∞

M

g′(z − s)

sα
ds
∣∣∣ . (2.3)

We estimate the first integral by taking the supremum in g′ and computing the remaining integral,
thus ∣∣∣ ∫ M

0

g′(z − s)

sα
ds
∣∣∣ ≤ A′

∫ M

0

ds

sα
=

A′

1− α
M1−α.

In the second integral, we first integrate by parts and pull out the supremum of g to deduce∣∣∣ ∫ ∞

M

g′(z − s)

sα
ds
∣∣∣ ≤ α

∣∣∣ ∫ ∞

M

g(z − s)

sα+1
ds
∣∣∣+AM−α ≤ 2AM−α.

Using these estimates in (2.3) gives

|Dα[g](z)| ≤ dα

( A′

1− α
M1−α + 2AM−α

)
. (2.4)

An easy computation shows that the minimum of the right-hand side of (2.4) is attained at
M = 2αA/A′ and this implies the first statement. The second is a consequence of the first by
taking the

supremum over all values in R. □

In some instances we shall need to split the integral operator (1.2) as follows

Dα[g](x) = dα

∫ x0

−∞

g′(y)

(x− y)α
dy + dα

∫ x

x0

g′(y)

(x− y)α
dy , for some x0 < x , (2.5)

and treat the first term as a known function, whereas the second one can be viewed as a left-sided
Caputo derivative, see e.g. [22], which we denote by Dα

x0
[·], indicating that the integration is from

a finite value x0, i.e. g ∈ C1
b ([x0,∞)) and α ∈ (0, 1].

Notice that the first term on the right-hand side of (2.5), which is a function of x, is not equal
to Dα[g](x0), which is a number for fixed x0. We shall use the following technical lemma several
times.

Lemma 2.2. For all ϕ ≤ −ϕ−(< 0), and defining H(ϕ) =
∫ ϕ

0
h(y)dy, we have

2ϕ3 ≤ h(ϕ) < Chϕ
3(< 0) and (0 <)H(ϕ)−H(ϕ−) < CHϕ

4

where

0 < Ch ≤ −2(ϕ− + ϕ+)ϕ+
(ϕ−)2

(< 1) and CH ≤ 2 .



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 7

The proof is an elementary calculus exercise. It is important to recall that constants here only
depend on ϕ+ and ϕ−.

2.2. Existence of trajectories that satisfy (1.6) and the derivation of (1.14). In this section
we prove the existence of solutions of (1.5) that satisfy (1.6) as ξ → −∞. Next, we prove that
there are three possible behaviours of such trajectories as ξ becomes large. The existence of these
trajectories follows directly from the results of [1], this means that we shall not need to prove some
steps, although we recall their proofs for completeness.

Namely, one obtains the following theorem by a direct application of the previous results and
a soft argument for the unbounded case.

Theorem 2.3. Given τ > 0, ϕ− and ϕ+ ∈ R such that (1.11) with ϕc = −(ϕ+ + ϕ−) < 0 holds.
Then,

(1) There exists a solution ϕ ∈ C3(−∞, 0) of (1.4) satisfying

lim
ξ→−∞

ϕ(ξ) = ϕ−

and ϕ′(ξ) < 0, for all ξ ∈ (−∞, 0), that is unique (up to a shift in ξ) among all ϕ ∈
ϕ− +H2(−∞, 0) ∩ C3

b (−∞, 0).
(2) Such solutions satisfy ϕ(ξ) < ϕ− for all ξ in the interval of existence.
(3) If such solutions are uniformly bounded, they exist for all ξ ∈ R and limξ→∞ ϕ(ξ) ∈

{ϕ+, ϕc}. Otherwise, there exists a finite ξ∗ ∈ R such that limξ→(ξ∗)− ϕ(ξ) = −∞.

The proofs of (i), (ii) and the first part of (iii) are a consequence of the results in [1]. We
recall some of the steps of the proof for these parts below. Thus, it remains to prove the second
statement of (iii): that unbounded solutions cannot be extended to the whole R. In the latter
case we first exclude the oscillatory behaviour (Lemma 2.7 below), i.e. we show that the limit is
−∞, and then we prove that this limit is reached at a finite value of ξ (Lemma 2.8).

Let us first summarise the implications of the results from [1] in the proof of Theorem 2.3. For

(i), one shows a ‘local’ existence result [1, Lemma 2] on (−∞, ξ̃] with ξ̃ < 0 and |ξ̃| sufficiently
large, that is based on linearisation about ϕ = ϕ− as ξ = −∞. For τ ≥ 0, all solutions of the
linearised equation

τv′′ +Dα[v] = h′(ϕ−)v , (2.6)

in the spaces Hs(−∞, ξ̃) with s ≥ 2 are of the form v(ξ) = beλξ, b ∈ R, where λ is the only real
and positive root of

P (z) = τz2 + zα − h′(ϕ−) , (2.7)

(see Appendix 7, Lemma 7.1). The statement can be proved as in [11], where a genuinely non-
linear flux function has been considered. The requirement in this proof is only to have h′(ϕ−) > 0,
which is guaranteed by (1.12). Then we can construct solutions of the non-linear problem (1.5)

on (−∞, ξ̃] as small perturbations of the exponential solutions of (2.6) as in [1]. The next step is
to extend for increasing ξ these solutions by a continuation principle [1, Lemma 3] and show that
there is uniqueness up to translation in ξ [1, Lemma 5].

Statement (ii) follows by the same arguments as in the proof of [1, Lemma 4]. Then the first
statement of (iii) is a direct consequence of [1, Lemma 6] that guarantees that under the given
assumption, the value of the limit must be a zero of h different from ϕ−. We recall that for the
quadratic case, h has only two zeros, ϕ− and ϕ+ (that would correspond to ϕc given the condition
(1.12)), but in the current case h has three zeros. The argument in the proof of [1, Lemma 6] is
by contradiction, assuming that the constant value of the limit of ϕ is not a zero of h, hence we
obtain the conclusion in Theorem 2.3 allowing the third possibility, ϕ+.

As a final remark, let us mention a crucial difference for the cubic flux. For the quadratic flux
one can show that solutions remain bounded and that this implies the existence of a limit value
as ξ → ∞ and therefore the only possible connection is to the constant value ϕ+, and this implies
the existence (and uniqueness up to translation in ξ). However, in the cubic case we cannot show



8 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

that solutions of (1.5) subject to (1.6) remain bounded from below. The main difference in the
arguments comes from the functional

H(ϕ) :=

∫ ϕ

0

h(y) dy = −cϕ
2

2
+
ϕ4

4
+Aϕ , with A = cϕ− − ϕ3− . (2.8)

The difference H(ϕ) − H(ϕ−) being non-negative is a necessary condition for existence (see
Lemma 2.4 below). In the quadratic case (and, more generally, for a genuinely non-linear flux) the
corresponding primitive has a zero ϕ̄ < ϕ+ that gives a lower bound of the solutions, because in the
interval (ϕ+, ϕ−), H(ϕ)−H(ϕ−) > 0. In the current case this is no longer true; H(ϕ)−H(ϕ−) > 0
is satisfied in (−∞, ϕ−), then, the existence of a lower bound is not guaranteed by this argument.
In the next lemmas we summarise the conditions on H.

Lemma 2.4. Let ϕ be a solution of (1.5) that satisfies (1.6) and let (−∞, ξexist) be its interval
of existence, where ξexist ∈ R ∪ {+∞}. Then∫ ξ

−∞
ϕ′(y)Dα[ϕ](y) dy ≥ 0 , ∀ξ ∈ (−∞, ξexist).

Moreover, the integral vanishes if and only if ϕ ≡ ϕ−.
As a consequence, for all ξ ∈ (−∞, ξexist), the following holds

0 ≤ τ

2
(ϕ′(ξ))2 +

∫ ξ

−∞
ϕ′(y)Dα[ϕ](y) dy = H(ϕ(ξ))−H(ϕ−) . (2.9)

The first statement appears in the proof of [1, Lemma 4] (see also [11] for a similar result) and
adapts the arguments of [26, Chapter 9]. The second part of the lemma follows by multiplying
(1.5) by ϕ′ and integrating with respect to ξ. The condition (2.9) is on the primitive of the
corresponding non-linear function of the equation, here (2.8).

Next, we show (1.14), which is a necessary condition on the far-field values for a solution of
(1.5)-(1.7) to exist.

Lemma 2.5. Let ϕ− and ϕ+ satisfy (1.11). Then, the inequality

H(ϕ+)−H(ϕ−) > 0 (2.10)

is a necessary condition to obtain a global solution ϕ of (1.5) that satisfies both (1.6) and (1.7).
Moreover, (2.10) is equivalent to

c < ϕ2− which implies ϕ− + ϕ+ > 0 . (2.11)

Proof. Suppose that ϕ is a global solution of (1.5) satisfying (1.6) and (1.7). Then (2.9) holds,
and taking the limit ξ → ∞ yields

H(ϕ+)−H(ϕ−) =

∫
R
ϕ′(ξ)Dα[ϕ](ξ) dξ ≥ 0.

We observe that H(ϕ) > H(ϕ−) for all ϕ ̸= ϕ−, and in particular (2.10) holds by (1.11). The
assertion (2.11) follows from (2.10) by elementary computations. □

Remark 2.6. We remark that if condition (2.10) does not hold then, Theorem 2.3 (ii) and (iii)
and Lemma 2.4 imply that the only possibility is that a solution that satisfies (i) of this theorem
is the one that has limξ→∞ ϕ(ξ) = ϕc. This is because it is the only limiting value, of the
three possibilities given in Theorem 2.3, that satisfies H(ϕc) − H(ϕ−) > 0 (see (2.9)). For the
same reason, there are no solitary waves supported by this model, since that would imply that
H(ϕ−)−H(ϕ−) = 0 and by Lemma 2.4 this means that ϕ must be constant.

Next we show that if a trajectory ϕ that satisfies (1.6) becomes unbounded, then it cannot
oscillate below a certain value.

Lemma 2.7 (Non-oscillatory behaviour). Let ϕ ∈ C3
b (−∞, 0) be a solution as constructed in

Theorem 2.3 (i)-(ii). If the continuation of ϕ becomes unbounded, then there exists ξ∗ ∈ R∪{+∞}
such that limξ→(ξ∗)− ϕ(ξ) = −∞.



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 9

Proof. Since ϕ(ξ) < ϕ− for all ξ ∈ R and, by assumption, ϕ is unbounded, there must exists
ξ∗ ∈ R ∪ {+∞} such that lim inf

ξ→(ξ∗)−
ϕ(ξ) = −∞.

We have to prove that lim
ξ→(ξ∗)−

ϕ(ξ) = −∞. We argue by contradiction, and assume that

limξ→(ξ∗)− ϕ(ξ) ̸= −∞ and does not exist. Then, by regularity, ϕ becomes unbounded in an oscil-
latory fashion as ξ increases. This means that there exists a sequence of local minima as follows:
{ξnmin}n≥0 is increasing and satisfies ξnmin → (ξ∗)−, ϕ′(ξnmin) = 0, ϕ′′(ξnmin) > 0 and ϕ(ξnmin) < −ϕ−
for all n ≥ 0 and {ϕ(ξnmin)}n≥0 is a monotone decreasing sequence with limn→∞ ϕ(ξnmin) = −∞.

Observe that also h(ϕ(ξnmin)) < 0 for all n ≥ 0, and this gives

Dα[ϕ](ξnmin) = h(ϕ(ξnmin))− τϕ′′(ξnmin) < 0 for all n ∈ N. (2.12)

Lemma 2.1 gives a bound for the fractional derivative in terms of ϕ and its first derivative.
Namely, there exists Cα > 0 (independent of τ) such that

|Dα[ϕ](ξ)| ≤ Cα∥ϕ∥1−α
L∞(−∞,ξ)∥ϕ

′∥αL∞(−∞,ξ) for all ξ ∈ (−∞, ξ∗)

and, in particular, for each ξ = ξnmin we obtain the lower bound

0 > Dα[ϕ](ξnmin) ≥ −Cα∥ϕ∥1−α
L∞(−∞,ξnmin)

∥ϕ′∥αL∞(−∞,ξnmin)
. (2.13)

On the other hand, considering Lemma 2.2 and that ϕ′′(ξnmin) > 0, we obtain the upper bound

Dα[ϕ](ξnmin) < h(ϕ(ξnmin)) < −Ch|ϕ(ξnmin)|3 = −Ch∥ϕ∥3L∞(−∞,ξnmin)
. (2.14)

Now, combining (2.13) and (2.14), we obtain

−Cα∥ϕ∥1−α
L∞(−∞,ξnmin)

∥ϕ′∥αL∞(−∞,ξnmin)
≤ −Ch∥ϕ∥3L∞(−∞,ξnmin)

which is equivalent to

∥ϕ∥α+2
L∞(−∞,ξnmin)

≤ Cα

Ch
∥ϕ′∥αL∞(−∞,ξnmin)

. (2.15)

We obtain an upper bound on ∥ϕ′∥L∞(−∞,ξnmin)
using Lemma 2.4 and Lemma 2.2, as follows:

τ

2
(ϕ′(ξ))

2 ≤ H(ϕ(ξ))−H(ϕ−) ≤ CHϕ
4(ξ) ≤ 2 (ϕ(ξnmin)))

4
, ∀ξ ∈ (−∞, ξnmin)

which implies

∥ϕ′∥2L∞(−∞,ξnmin)
≤ 4

τ
∥ϕ∥4L∞(−∞,ξnmin)

. (2.16)

Finally, combining (2.15) and (2.16) implies for all n ∈ N that

∥ϕ∥2−α
L∞(−∞,ξnmin)

< τ−α/2C,

with C = 2α Cα

Ch
. But this contradicts that limn→∞ ϕ(ξnmin) = −∞ and such sequences of local

minima cannot exists. Thus it must be that limξ→(ξ∗)− ϕ(ξ) = −∞. □

Next we show that ξ∗ of the previous lemma is a finite value.

Lemma 2.8. Let ϕ ∈ C3
b (−∞, 0) be a solution as constructed in Theorem 2.3 (i)-(ii). If the

continuation of ϕ becomes unbounded, then there exists a finite ξ∗ ∈ R such that

lim
ξ→(ξ∗)−

ϕ(ξ) = −∞, (2.17)

and, therefore, ϕ cannot be extended to R. Moreover, the asymptotic behaviour of ϕ is given by

lim
ξ→(ξ∗)−

|ϕ(ξ)| (ξ∗ − ξ) =
√
τC (2.18)

where C > 0 is a constant independent of τ .



10 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

Proof. Since ϕ is unbounded, by Lemma 2.7 there exist ξ∗ ∈ R ∪ {+∞} and ξ1 ∈ R such that

lim
ξ→(ξ∗)−

ϕ(ξ) = −∞ and ϕ′(ξ) < 0, ∀ξ ∈ (ξ1, ξ
∗). (2.19)

It is now convenient to rewrite the equation (1.5), as a first order system by making the change
of variables u(ξ) = ϕ(ξ) and v(ξ) = ϕ′(ξ) for ξ ∈ (−∞, ξ∗). This gives

u′ = v,

v′ =
1

τ
(h(u)−Dα[u]) .

(2.20)

We first notice that there exist some ξ0 ∈ [ξ1, ξ
∗) and Cv > 0 such that

−∞ < u(ξ) < −ϕ− ∀ξ ∈ [ξ0, ξ
∗), u(ξ0) ≤ u(ξ) < ϕ− ∀ξ ≤ ξ0 (2.21)

and

v(ξ) < −Cv, ∀ξ ∈ (ξ0, ξ
∗). (2.22)

The bounds (2.21) hold by (2.19). Let us show (2.22): If v becomes unbounded there is nothing
to prove, again by (2.19). On the other hand, if v is bounded, we obtain an upper bound of v′(ξ),
which diverges to −∞ as ξ → (ξ∗)−. Indeed, applying Lemma 2.1, Lemma 2.2 and an estimate
like (2.16) yield

v′(ξ) =
1

τ
h(u(ξ))− 1

τ
Dα[u](ξ)

≤ Ch

τ
u(ξ)3 +

2αCα

τ1+α/2
|u(ξ)|1+α

= u(ξ)3
(Ch

τ
− 2αCα

τ1+α/2

1

|u(ξ)|2−α

)
.

The right-hand side of this inequality tends to−∞ as ξ → (ξ∗)−, therefore, limξ→(ξ∗)− v
′(ξ) = −∞.

This implies (2.22). We can adjust the value of ξ0 by taking it closer to ξ∗ as necessary so that
both bounds hold in the same interval.

Once (2.21) and (2.22) are established for ξ ∈ (ξ0, ξ
∗), we introduce the variables

z :=
1

u
< 0 and w := − v

u2
≥ 0

in such interval.
We also change the independent variable, in order to absorb z in the derivative, as follows,

s = s0 −
∫ ξ

ξ0

dy

z(y)
, s0 > 0. (2.23)

Notice that z < 0, thus s is strictly increasing with respect to ξ.
The system (2.20) then reads

dz

ds
= −zw,

dw

ds
= −2w2 +

1

τ

(
h
(1
z

)
−Dα

[1
z

])
z3.

(2.24)

Case I: We first analyse the possibilities of ‘extinction’ and of ‘blow-up’ for w at a finite s. Let
us assume the former: there exists a finite s̄ > s0 such that lims→s̄ w(s) = 0. Then, at ξ̄ ≤ ξ∗,

given by s̄ = s0 −
∫ ξ̄

ξ0
u(y)dy, either v(ξ̄) = 0 or limξ→(ξ̄)− u(ξ) = −∞. The former contradicts

(2.22). The latter case implies that either s̄ is infinite, which gives a contradiction, or that u is
integrable in (ξ0, ξ̄). In that case, we have that ξ̄ = ξ∗ < ∞, but also that for ξ ∈ (ξ0, ξ

∗) there
are very small constants ε1 > 0 and ε2 > 0, such that

ε1 < − u′(ξ)

u(ξ)2
< ε2 (2.25)



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 11

and integrating gives

1

ε2(ξ − ξ0) + 1/u(ξ0)
< u(ξ) <

1

ε1(ξ − ξ0) + 1/u(ξ0)

so that ξ∗ ≤ ξ0 − ε1
u(ξ0)

. The constant ε1 tends to 0 as ξ → (ξ∗)− (because we have only used the

definition of w and the limit w → 0 in the estimate (2.25)), implying that ξ∗ = ξ0, but this gives
a contradiction, with the choice of ξ0. This implies that w cannot tend to 0 at a finite value.

Let us assume now that w exhibits ‘blow-up’. Then there exists a finite s∗ > s0 such that
lims→s∗ w(s) = ∞. Thus, for ξ ∈ (ξ0, ξ

∗) sufficiently close to ξ∗,

− u′(ξ)

u2(ξ)
≫ 1

and this, integrating over (ξ0, ξ), gives

u(ξ) <
1

ξ − ξ0 + 1/u(ξ0)
,

but, this implies that s∗ cannot be finite (using the above bound in the integral of (2.23) gives ∞
as lower bound) and w does not blow up.

Case II: Let us now assume that w is defined for all s ∈ R. We shall show that ∃ lims→∞ w(s) <∞,
and then apply the definition of the new variables and integrate to get the result. First, we obtain
estimates from (2.24).

Integrating the first equation in (2.24), we obtain

z(s) = z0e
−

∫ s
s0

w(s) ds → 0− as s→ ∞. (2.26)

This limit is clear if
∫∞
s0
w(s) ds = +∞. If

∫∞
s0
w(s) ds < +∞, in particular lims→∞ w(s) = 0, then

an argument as for the extinction of w gives that either lims→∞ v(s) = 0, which is in contradiction
with (2.22), or that lims→∞ u(s) = −∞, which also implies that z(s) → 0 as s→ ∞.

Now, we observe that Dα[u] < 0 for all ξ ∈ (ξ0, ξ
∗). To prove this we first split Dα[u](ξ) as

follows,
1

dα
Dα[u](ξ) =

∫ ξ0

−∞

v(y)

(ξ − y)α
dy +

∫ ξ

ξ0

v(y)

(ξ − y)α
dy.

The second integral is negative because of (2.22). And the first is also negative, since integrating
by parts and using (2.21) we obtain∫ ξ0

−∞

v(y)

(ξ − y)α
dy = −α

∫ ξ0

−∞

u(y)

(ξ − y)α+1
dy +

u(ξ0)

(ξ − ξ0)α

< α|u(ξ0)|
∫ ξ0

−∞

dy

(ξ − y)α+1
+

u(ξ0)

(ξ − ξ0)α
= 0.

Let us now obtain a lower bound for Dα[1/z] = Dα[u] by rewriting Lemma 2.1 in terms of z
and w. We have two cases for all s > s1 for some s1 ≥ s0 large enough:

0 < −Dα
[1
z

]
≤ D(z, w) (2.27)

with

D(z, w) :=

{
C ′

α| 1z |
1−α, if |v| stays bounded,

Cα| 1z |
1+α(sups>s1 w)

α, if |v| becomes unbounded,

for some C ′
α, Cα > 0.

We now obtain bounds for the non-linear term using (2.26). Let s2 ≥ s0 be large enough such
that for all s > s2,

|z(s)| < min
{ c

−(cϕ− − ϕ3−)
,
1√
c

}
,

then

0 <
1

τ
− c

τ
z2 ≤ 1

τ
z3h

(1
z

)
=

1

τ
− 1

τ
(c− (cϕ− − ϕ3−)z)z

2 ≤ 1

τ
for all s > s2. (2.28)



12 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

From (2.27) and (2.28) we have the following bounds on dw/ds using the second equation in (2.24):

1

τ
− c

τ
z2 − 2w2 − 1

τ
(−z)3D(z, w) ≤ dw

ds
≤ 1

τ
− 2w2, ∀s > max{s1, s2}. (2.29)

To prove that w → C > 0 as s → ∞, we now argue by contradiction. First, we assume that
w becomes unbounded, then (2.29) implies that w is decreasing as long as w > 1/

√
2τ , but this

contradicts that 0 < w(s) becomes unbounded. Now, we assume that the limit is finite with
C = 0, since also lims→∞ z(s) = 0, these implies that dw(s)/ds > 0 for all s large enough. This
contradicts that C = 0 and w(s) > 0.

The last possibility that we have to exclude is that w oscillates without limit. Let M =
sups>max{s1,s2} w(s) < ∞, then using (2.29) we obtain there exists C > 0 and s3 ≥ max{s1, s2}
such that

1

τ
− 2w(s)2 − C|z(s)|2−α ≤ dw(s)

ds
≤ 1

τ
− 2w(s)2 for all s > s3. (2.30)

Here we have used (2.27) and noticed that for |z| small enough we have |z|2+α, |z|2 < |z|2−α.

Observe that if M < 1/
√
2τ , no oscillations are possible in the limit, because there exists

s4 ≥ s3 such that

0 <
1

τ
− 2M2 − C|z(s)|2−α ≤ dw(s)

ds
for all s > s4.

Then, in this case, we obtain the desired result: lims→∞ w(s) =M > 0 and is finite.

Now, if M = 1/
√
2τ , there exists s ∈ R such that w(s) < 1/

√
2τ , because we are assuming that

the limit of w does not exist. If M > 1/
√
2τ , then by (2.30), with s > s3,

dw(s)

ds
≤ 1

τ
− 2w(s)2 ≤ 0, as long as

1√
2τ

≤ w(s) ≤M.

This means that on the intervals of s for which 1√
2τ

≤ w(s) ≤M , w is not increasing, so w cannot

oscillate in this range. This implies that w(s) ∈ (0, 1/
√
2τ ], for all s > s4 with s4 > s3 large

enough, oscillating without limit.
Then, there exists also an increasing sequence {sn}n≥0 > s4, where local minima of w are

attained, with dw(sn)/ds = 0, 0 < w(sn) ≤ 1/
√
2τ and sn → ∞ as n → ∞. Let the sequence

{δn}n≥0 be defined by evaluating the right-hand side of (2.30) at each s = sn,

0 < δn :=
1

τ
− 2w2(sn) <

1

τ
.

Again there are two possibilities: either {δn}n≥0 is bounded from below by a positive constant K,
or δn → 0 as n→ ∞. In the former case, from (2.30) we obtain that for all n,

K − C|z(sn)|2−α ≤ dw

ds
(sn) = 0.

Applying (2.26) as n → ∞, we deduce that there exists n0 ≥ 0 such that the left hand side is
strictly positive for all n ≥ n0, a contradiction.

If δn → 0 as n → ∞, then w(sn) → 1/
√
2τ as n → ∞. This means that this sequence of

local minima is converging to the supremum of w on s > s4, but this contradicts that w oscillates
without a limit and, moreover, this implies that lims→∞ w(s) = 1/

√
2τ .

With the limit of w and taking into account its regularity, we obtain that there exists positive
constants C1 and C2 proportional to 1/

√
τ , such that for all ξ > ξ0,

C1 < − u′(ξ)

u2(ξ)
< C2.

Then, integrating over the interval (ξ0, ξ) for ξ < ξ∗, gives

1

C2(ξ − ξ0) + 1/u(ξ0)
< u(ξ) <

1

C1(ξ − ξ0) + 1/u(ξ0)
.

These bounds imply (2.18), since C1 and C2 are proportional to 1/
√
τ , and with ξ∗ ≤ ξ0−

√
τ

Cu(ξ0)
,

for some positive constant C. □



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 13

Gathering the results of the previous lemmas we finish this section.

Proof of (iii) of Theorem 2.3. As explained earlier the proofs of (i), (ii) and the first part of (iii)
follow from the results of [1]. The proof of the last part of (iii) follows applying the previous
lemmas. First, Lemma 2.7 rules out the oscillatory behaviour of ϕ and then Lemma 2.8 ensures
that (2.17) is satisfied. □

3. Existence of undercompressive waves: Proof of Theorem 1.1

In the forthcoming, for every value of τ > 0 we will let ϕτ (ξ) denote a solution of equation (1.5)
satisfying (1.6) as constructed in Theorem 2.3. According to the three possible behaviours of such
trajectories, established in Theorem 2.3 (iii), we define the following sets of τ ’s:

Definition 3.1. For every τ > 0 let ϕτ be a solution as constructed in Theorem 2.3. Then we
define the sets

Σu :=
{
τ > 0 : lim

ξ→(ξ∗)−
ϕτ (ξ) = −∞ for some ξ∗ ∈ R

}
,

Σc :=
{
τ > 0 : lim

ξ→∞
ϕτ (ξ) = ϕc

}
,

Σ+ :=
{
τ > 0 : lim

ξ→∞
ϕτ (ξ) = ϕ+

}
.

By the definition and uniqueness up to translation, these sets are disjoint. If the set Σ+ is non-
empty, then there exists a solution ϕτ of the problem (1.5)-(1.6), such that limξ→∞ ϕτ (ξ) = ϕ+,
thus showing that will complete the proof of Theorem 1.1.

We prove this by a shooting argument, where τ is the shooting parameter. We divide this
section into two. In the first part we show that Σu is non-empty and open and in the second we
show that Σc is non-empty and then also that Σ+ is non-empty. In this final part of the proof,
we argue by contradiction; we assume that Σ+ is empty, this means that Σc is closed, and using
continuity with respect to τ > 0, where we invoke Appendix 6, we obtain to a contradiction.

We remark that we use Theorem 6.1 of Appendix 6 for our problem, that is rewritten as (6.4)
with (6.5)-(6.7). This is valid on finite intervals, but, as we shall see, we can get continuity with
respect to τ > 0 on intervals (−∞, ξ) using the results of [1].

We notice that as a main step to show that Σc is non-empty we need to consider a modified
problem where the non-linearity has only the zeros ϕ− and ϕc and coincides with h in that range.
Then we can apply the monotonicity results of [1] and a new result, Theorem 3.5, that guarantees
that trajectories stay in a range where both non-linearities coincide.

3.1. The set Σu. We first show that Σu is non-empty:

Lemma 3.2. Consider ϕ− and ϕ+ satisfying (1.11) and (1.14). Let ϕτ denote the unique (up to
shifts in ξ) solution of (1.5) satisfying (1.6) as constructed in Theorem 2.3. Then, there exists
τm > 0 such that for all τ > τm there exists ξ∗τ ∈ R such that limξ→(ξ∗τ )

− ϕτ (ξ) = −∞.

Proof. Let us argue by contradiction. Assume that for all τ0 > 0 there exists at least one τ > τ0
such that ϕτ (ξ) is defined for all ξ ∈ R. By Theorem 2.3, we know that ϕτ is smooth, ϕτ (ξ) < ϕ−
for all ξ ∈ R, and limξ→∞ ϕτ (ξ) = ϕ∗ ∈ {ϕc, ϕ+}. Moreover,

−ϕ− < ϕ+ < ϕc < 0 (3.1)

because of (1.14), see also Lemma 2.5.
First, we prove that ∥ϕτ∥∞ = ϕ− and deduce estimates on ∥ϕ′τ∥∞ and on ∥Dα[ϕτ ]∥∞.
At this point, one has just a lower bound for ∥ϕτ∥∞ ≥ ϕ−. Then, we distinguish two cases:

Either infξ∈R ϕτ (ξ) = ϕ∗ or there exists a value ξmin ∈ R such that ϕτ (ξmin) = minξ∈R ϕτ (ξ) =:
ϕmin.

First, if infξ∈R ϕτ (ξ) = ϕ∗ then ∥ϕτ∥∞ ≤ max{|ϕ∗|, ϕ−} = ϕ− because of (3.1), and this implies
∥ϕτ∥∞ = ϕ− in this case.

In the other case, let ξmin ∈ R be the point at which the minimum of ϕτ is attained. It is
enough to prove that ϕτ (ξmin) =: ϕmin ∈ (−ϕ−, ϕc), since this implies that ∥ϕτ∥∞ ≤ ϕ−, because
of (3.1).



14 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

We argue by contradiction and assume to the contrary that ϕmin < −ϕ−. Now, using that
ϕ′′τ (ξmin) ≥ 0 in (1.5) yields

0 > h(ϕmin) = τϕ′′τ (ξmin) +Dα[ϕτ ](ξmin) ≥ Dα[ϕτ ](ξmin), (3.2)

and, by Lemma 2.1, there exists Cα > 0 (independent of τ) such that

|Dα[ϕτ ](ξ)| ≤ Cα∥ϕτ∥1−α
∞ ∥ϕ′τ∥α∞ for all ξ ∈ R. (3.3)

Then, combining (3.2) and (3.3), we conclude that

0 > Dα[ϕτ ](ξmin) ≥ −Cα∥ϕτ∥1−α
∞ ∥ϕ′τ∥α∞. (3.4)

Now, Lemma 2.2 implies that there exists a constant Ch > 0, depending only on ϕ− and ϕ+, such
that

Dα[ϕτ ](ξmin) ≤ h(ϕmin) < Chϕ
3
min = −Ch∥ϕτ∥3∞ < 0. (3.5)

Combining (3.4) and (3.5), then gives

∥ϕτ∥2+α
∞ ≤ Cα

Ch
∥ϕ′τ∥α∞ (3.6)

where the constants Cα and Ch depend on α, ϕ− and ϕ+ but are independent of τ .
On the other hand, Lemmas 2.4 and 2.2 imply that

τ

2
(ϕ′τ (ξ))

2 ≤ H(ϕτ (ξ))−H(ϕ−) ≤ 2∥ϕτ∥4∞ for all ξ ∈ R,

and taking the supremum with respect to ξ ∈ R yields
τ

2
∥ϕ′τ∥2∞ ≤ 2∥ϕτ∥4∞. (3.7)

Finally, with (3.6) and (3.7), we obtain

ϕ2−α
− < ∥ϕτ∥2−α

∞ < τ−α/2C, (3.8)

with C = 2αCα/Ch > 0, which is independent of τ . The hypothesis ϕ(ξmin) < −ϕ− implies

that the inequalities in (3.8) are strict, then, necessarily τ < C2/αϕ
2−4/α
− . Thus, for τ > τα :=

C2/αϕ
2−4/α
− the bounded solution ϕτ satisfies ∥ϕτ∥∞ = ϕ−.

Observe that the estimates (3.3) and (3.7) are valid in both cases considered above, then
substituting ∥ϕτ∥∞ = ϕ− in them we obtain

∥ϕ′τ∥∞ ≤ τ−1/22ϕ2−, and ∥Dα[ϕτ ]∥∞ ≤ τ−α/2Cα2
αϕα+1

− . (3.9)

We shall use these estimates below.
To complete the proof, we have to get a contradiction with the assumption that there are such

bounded solutions if τ is large enough. For the argument, we rescale the variables as follows

ξ =
√
τX and ϕτ (ξ) = Ψτ (X)

such that (1.5) reads
d2

dX2
Ψτ + τ−α/2Dα

X [Ψτ ] = h(Ψτ ). (3.10)

Then the estimates (3.9) induce the uniform bounds:

∃C > 0 (independent of τ) such that ∥Dα
X [Ψτ ]∥∞ < C, ∥Ψ′

τ∥∞ < C. (3.11)

By Theorem 2.3(i) and its proof, for sufficiently small ε > 0, there exists X0 ∈ R such that
Ψτ (X0) = ϕ− − ε and Ψ′

τ (X0) < 0 with ϕ− − ε > ϕm where ϕm ∈ (ϕc, ϕ−) such that h′(ϕm) = 0.
Let X1 ∈ R such that Ψτ (X1) ∈ (ϕc, ϕm) and h(Ψτ (X1)) = h(Ψτ (X0)). Choosing ε even smaller,
we can ensure that ϕc < Ψ(X1) < 0 < ϕm. Then, integrating (3.10) over the interval (X0, X) and
using (3.11) yields

Ψ′
τ (X) ≤ Ψ′

τ (X0) + τ−α/2C(X −X0) +

∫ X

X0

h(Ψτ (Y )) dY .

For all X ∈ (X0, X1), we deduce h(Ψτ (X)) ≤ h(Ψτ (X0)) = h(ϕ− − ε) < 0 and

Ψ′
τ (X) ≤ Ψ′

τ (X0) +
{
τ−α/2C + h(ϕ− − ε)

}
(X −X0).



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 15

Choosing τ0 > 0 sufficiently large, such that the associated τ > τ0 satisfies τ−α/2 < |h(ϕ−−ε)|
2C ,

implies that Ψ′
τ (X) < 0 for all X ∈ (X0, X1). And therefore, also, Ψτ decreases monotonically for

all X ∈ (−∞, X1) with ϕc < Ψτ (X1) < 0.
If Ψτ is not monotone for all X ∈ R then it attains its first local minimum at some Xmin > X1.
We now evaluate the energy estimate (2.9), rescaled as for (3.10), at Xmin and using the bound

(3.11) for Dα
X [·] yields

0 ≤ H(Ψτ (Xmin))−H(ϕ−) = τ−α/2

∫ Xmin

−∞
Ψ′

τ (Y )Dα
Y [Ψτ ] dY < τ−α/2C

∫ Xmin

−∞
|Ψ′

τ (Y )| dY.

Using that Ψτ is decreasing in (−∞, Xmin) and that ∥Ψτ∥∞ = ϕ−, implies

0 ≤ H(Ψτ (Xmin))−H(ϕ−) < τ−α/2C

∫ Xmin

−∞
|Ψ′

τ (Y )| dY

= τ−α/2C(ϕ− −Ψτ (Xmin)) < 2τ−α/2Cϕ−.

(3.12)

Observe that, H(Ψτ (Xmin))−H(ϕ−) ≥ H(ϕ+)−H(ϕ−) > 0, since Ψτ (Xmin) ≤ Ψτ (X1) < 0 and
by Lemma 2.5 (indeed, H(ϕ) − H(ϕ−) has two local minima, one at ϕ− which is zero, and the
other at ϕ+, which is strictly positive; at ϕc it attains a local maximum). On the other hand, the
upper bound in (3.12) can be made arbitrarily small by choosing τ0 sufficiently large. This gives
a contradiction, thus Ψτ does not attain a minimum and decreases for all X ∈ R.

We have thus concluded that the bounded solution Ψτ converges either to ϕ+ or ϕc in a
monotonically decreasing way. We can use the previous argument again and take the limit Xmin →
∞ in the energy estimate, this gives

0 < H(ϕ∗)−H(ϕ−) < τ−α/2C(ϕ− − ϕ∗) < 2τ−α/2Cϕ−.

However, 0 < H(ϕ+) − H(ϕ−) ≤ H(ϕc) − H(ϕ−) is a fixed positive number whereas the up-
per bound can be made arbitrarily small by choosing τ0 sufficiently large. This gives again the
contradiction, and so there cannot exist such bounded solutions if τ is large enough. □

Lemma 3.3. Σu is an open set.

Proof. By Lemma 3.2, there exists a value τm > 0 such that (τm,+∞) ⊂ Σu, thus such points are
inner points of Σu. Then, it remains to prove that if the intersection (0, τm] ∩ Σu is non-empty
then is an open set.

Suppose τ0 ∈ (0, τm] ∩ Σu, in particular limξ→(ξ∗τ0
)− ϕτ0(ξ) = −∞ for some ξ∗τ0 . We next prove

that there exists δ > 0 such that for all τ ∈ (τ0 − δ, τ0 + δ), the solution ϕτ of (1.5) and (1.6)
satisfies limξ→(ξ∗τ )

− ϕτ = −∞ for some ξ∗τ ∈ R.
We use continuous dependence on the parameter τ on finite intervals (see Appendix 6). Given

a bounded interval I such that ϕτ0(ξ) < −ϕ− for all ξ ∈ I, then for all ε > 0 there exists δ > 0
such that

|ϕτ0(ξ)− ϕτ (ξ)| < ε, for ξ ∈ I and τ ∈ (τ0 − δ, τ0 + δ). (3.13)

Let C = 2αCα/Ch, as in the proof of Lemma 3.2, and ε > 0 fixed. We take

ξ ≥ ξsup := sup
{
ξ ∈ (−∞, ξ∗τ0) : ϕτ0(ξ) = −(max{ϕ−, (τ

−α
2

0 C)
1

2−α }+ ε)
}

and choose a bounded interval I by means of,

I :=
{
ξ ∈ (ξsup, ξ

∗
τ0) :

− (2max{ϕ−, (τ
−α

2
0 C)

1
2−α }+ ε) < ϕτ0(ξ) < −(max{ϕ−, (τ

−α
2

0 C)
1

2−α }+ ε)
}
.

Thus, there exists δ > 0 such that (3.13) holds. Then, there we take (a maybe smaller value than
the previous) δ such that δ < (1− 2−2/α)τ0, and we define a sub-interval J ⊆ I such that

ϕτ0(ξ) < −(max{ϕ−, ((τ0 − δ)−
α
2 C)

1
2−α }+ ε) for all ξ ∈ J.

Then, since |ϕτ0(ξ)| − |ϕτ (ξ)| < ε by (3.13),

|ϕτ (ξ)| > |ϕτ0(ξ)| − ε > max{ϕ−, ((τ0 − δ)−α/2C)
1

2−α } (3.14)



16 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

for all ξ ∈ J and τ ∈ (τ0 − δ, τ0 + δ) for δ sufficiently small.
Let us now argue by contradiction and suppose that there exists some τ ∈ (τ0 − δ, τ0 + δ) such

that τ /∈ Σu. Then this means that ϕτ is bounded and one can apply the first part of the proof of
Lemma 3.2, see (3.8), and obtain

∥ϕτ∥∞ ≤ (τ−α/2C)
1

2−α ,

where C as above. We recall that to get to this inequality the starting assumption is that ϕτ has
its minimum in the interval (−∞,−ϕ−). Now, the inequality contradicts (3.14), since τ0 − δ < τ .
Therefore, (τ0−δ, τ0+δ) ⊂ Σu and we have that Σu is open with the usual topology in (0,+∞). □

3.2. The sets Σc and Σ+: end of the proof. In this section we complete the proof of Theo-
rem 1.1.

Before we state the result that we need, which is proved in Section 4, let us introduce the
modified problem:

τϕ′′τ +Dα[ϕτ ] = h̃(ϕτ ) , (3.15)

with

h̃(ϕ) :=

{
h(ϕ), if ϕ ≥ −

√
c/3 ,

Pc(ϕ), if ϕ ≤ −
√
c/3 ,

(3.16)

where Pc(ϕ) is a function such that Pc(−
√
c/3) = h(−

√
c/3), P ′

c(−
√
c/3) = h′(−

√
c/3) = 0 and

P ′′
c (−

√
c/3) = h′′(−

√
c/3), and such that Pc(ϕ) > 0 for all ϕ ≤ −

√
c/3. For example, we can

choose

Pc(ϕ) = Aϕ4 +Bϕ3 + Cϕ2 +Dϕ+ E

such that A > 0 and the rest of coefficients are chosen such that, Pc(−
√
c/3) = h(−

√
c/3),

P ′
c(−

√
c/3) = h′(−

√
c/3) = 0 and P ′′

c (−
√
c/3) = h′′(−

√
c/3) < −6

√
c/3 < 0 (these give C and

D as a linear combination of A and B), and such that the local minimum at some ϕmin < −
√
c/3

(this gives a linear relation for A and B, and choosing B very negative guarantees that A is
positive) has Pc(ϕmin) > 0 (this is achieved by taking E > 0 as large as necessary). This last

condition guarantees that h̃(ϕ) > 0 for all ϕ ≤ −
√
c/3.

Observe that at ϕ = −
√
c/3, h attains its local maximum and that −

√
c/3 < ϕc < 0, thus this

modification of h is C2 at the point −
√
c/3. We observe that we also denote by ϕτ a solution to

this problem (for simplicity of notation), but it will be made clear to which problem such profile
corresponds.

We define, analogously to (2.8), the primitive of h̃, H̃(ϕ) =
∫ ϕ

0
h̃(y)dy. We observe that h̃ has

only two zeros, namely, ϕc and ϕ−, and the necessary condition, see Lemma 2.4 (2.9),

0 ≤ H̃(ϕ)− H̃(ϕ−) =

∫ ϕ

ϕ−

h̃(y) dy (3.17)

holds only for ϕ ∈ [ϕ̄, ϕ−], where ϕ̄ is the zero of H̃(ϕ) − H̃(ϕ−) that satisfies ϕ̄ < ϕc. One can
then adapt the results of [1] to this equation subject to the far-field behaviour

lim
ξ→−∞

ϕτ (ξ) = ϕ− and lim
ξ→+∞

ϕτ (ξ) = ϕc. (3.18)

In fact, solutions to this problem exist for all τ > 0 and lie in the interval (ϕ̄, ϕ−). We also recall
that ϕ− and ϕc satisfy the Rankine-Hugoniot condition, giving the same wave speed c as in (1.8),
because ϕc is also a root of h.

We shall denote by ϕ0 the solutions of the equation with τ = 0:

Dα[ϕ0] = h̃(ϕ0) , (3.19)

where h̃ is given in (3.16) such that

lim
ξ→−∞

ϕ0(ξ) = ϕ− and lim
ξ→+∞

ϕ0(ξ) = ϕc. (3.20)

These solutions are constructed as in [2] (see also [4]) and are monotone decreasing.



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 17

With regard to the monotonicity analysis, we recall the following result of [1], that concludes
monotonicity for τ small enough in a large interval, by comparing the solutions of (3.15)-(3.18)
for τ > 0 small to the solution of (3.19)-(3.20).

Theorem 3.4 ([1, Theorem 9]). Let ϕτ be a solution of (3.15)-(3.18). If τ is sufficiently small,

then there exists an interval Iτ = (−∞, ξτ ] with ξτ = O(τ−
1

2−α ) as τ → 0+, and a value ξ = ξ0τ <
ξτ such that ϕτ (ξ

0
τ ) = ϕ0(ξ

0
τ ), moreover, |ϕτ (ξ)−ϕ0(ξ)| ≤ τC and |ϕ′τ (ξ)−ϕ′0(ξ)| ≤ τ1/(2−α)C for

all ξ ∈ Iτ . Thus for τ sufficiently small, ϕτ is also monotone decreasing in Iτ .

The proof of this result can be adapted to the travelling wave equation (3.15), without change,
since it only requires the construction of solutions on intervals of the form (−∞, ξ] and a uniform
bound on the solution.

From this result we have conjectured monotonicity in the ‘tail’ as well (see [1]) for sufficiently
small τ . This is still an open problem, but the conjecture is supported by the numerical compu-
tations in [12]. Instead of monotonicity, we prove the following theorem, that is sufficient for our
purpose.

Theorem 3.5. Let ϕτ be a solution of (3.15)-(3.18). If τ > 0 is sufficiently small, then there exists

a constant Cτ = O(τ
α

2−α ) as τ → 0+, such that −
√
c/3 < ϕc−Cτ < ϕ− and ϕτ (ξ) ∈ (ϕc−Cτ , ϕ−)

for all ξ ∈ [ξτ ,∞), where ξτ = O(τ−
1

2−α ) as τ → 0+ as in Theorem 3.4.

As we have mentioned earlier, we postpone the proof of this result to Section 4. We assume for
the rest of this section that the result is true. With that we can prove the following Lemma.

Lemma 3.6. Let (ϕ−, ϕ+; c) satisfy the Rankine-Hugoniot condition (1.8) and (1.11) with ϕc =
−ϕ− − ϕ+. If τ > 0 is sufficiently small then there exists a solution ϕτ ∈ C3

b (R) of the problem
(1.5) and (3.18). It is unique (up to a shift) among all ϕτ ∈ ϕ− + H2(−∞, 0) ∩ C3

b (R). In
particular, this means that the set Σc is non-empty.

Proof. If τ > 0, the existence of solutions to (3.15)-(3.18) is shown as in [1]. We now use The-
orem 3.4 and Theorem 3.5, in particular, for such sufficiently small τ , these solutions satisfy
−
√
c/3 < ϕτ (ξ) < ϕ− for all ξ ∈ R. In this range of ϕ’s, h and h̃ coincide, thus, these solutions

are also solutions of the original equation (1.5), since they satisfy (3.18), and this finishes the
proof. □

We are in the position to complete the proof of our main result.

Proof of Theorem 1.1. Throughout this proof, C denotes a positive constant of order 1 which is
independent of τ and of any other parameter or small constant that may be used.

We have to prove that Σ+ is non-empty. We argue by contradiction; let us assume that Σ+ = ∅.
Now, by Theorem 2.3, we have that Σ+, Σu and Σc are disjoint, and that Σ+∪Σu∪Σc = (0,+∞).
From Lemmas 3.2 , 3.3 and 3.6, we then deduce that the set Σc must be closed in (0,+∞). This
implies the existence of a τ0 ∈ Σc ∩ ∂Σc such that, for all δ > 0 the set Σu ∩ (τ0 − δ, τ0 + δ) is
non-empty.

Since τ0 ∈ Σc, the far-field behaviour of ϕτ0 is given by (3.18). We recall the construction
of solutions (see [1, Lemma 2]) satisfying (1.6): For all τ > 0, let λτ be the positive root of
τz2 + zα − h′(ϕ−) = 0 and for any ε > 0 let also Iτ,ε = (−∞, ξτ,ε] with ξτ,ε = log ε/λτ . Then,
there exists an order one constant C > 0, such that

ϕτ ∈ ϕ− +H2(Iτ,ε), ∥ϕτ − ϕ− − eλτξ∥H2(Iε) ≤ Cε2 .

We conclude, that for all ε > 0 there exists δε > 0 and ξε, defined by

ξε = inf
τ∈(τ0−δε,τ0+δε)

{ log ε

λτ

}
,

such that, for all τ ∈ (τ0 − δε, τ0 + δε),

|ϕτ0(ξ)− ϕτ (ξ)|, |ϕ′τ0(ξ)− ϕ′τ (ξ)| < ε, ∀ξ < ξε.

At this stage, we can apply Theorem 6.1, as explained in Appendix 6, about continuous depen-
dence on finite intervals, to get possibly a smaller neighbourhood of τ0 for which solutions and



18 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

their first derivative remain close, by an order ε constant, to ϕτ0 and its derivative, respectively,
in a much larger interval (−∞, ξετ0 ] with ξ

ε
τ0 > ξε.

Now, we follow the steps of Appendix 8 to solve the equation implicitly as in (8.6) for a =
−h′(ϕc) > 0. For convenience we also introduce the notation

h′c := h′(ϕc) < 0 . (3.21)

We split the interval of integration on the non-local operator Dα[·] at ξετ0 . Thus, the integral
part for ξ > ξετ0 can be considered as a classical Caputo derivative, while the remainder is treated
as a known inhomogeneity. This gives

τ0ϕ
′′
τ0 +Dα

ξετ0
[ϕτ0 ]− h′cϕτ0 = h(ϕτ0)− h′cϕτ0 − dα

∫ ξετ0

−∞

ϕ′τ0(y)

(ξ − y)α
dy . (3.22)

For convenience, we can shift the independent variable as η = ξ−ξετ0 and introduce the function

Φτ0(η) = ϕτ0(ξ)− ϕc.

Applying these changes of variables to (3.22) gives the equation

τ0Φ
′′
τ0(η) +Dα

0 [Φτ0(η)]− h′cΦτ0(η) = Q(η; τ0)

where Q is the term that contains the remainder terms of the new formulation, namely,

Q(η; τ0) := h(ϕτ0(η + ξτ0))− h′cΦτ0(η)− dα

∫ 0

−∞

Φ′
τ0(z)

(η − z)α
dz.

We can rewrite this term, using the explicit formula (1.5) of h, as follows (we shall use this notation
below)

Q(η; τ0) := (Φτ0(η) + 3ϕc)(Φτ0(η))
2 +R(η; τ0) (3.23)

with

R(η; τ0) = −dα
∫ 0

−∞

Φ′
τ0(z)

(η − z)α
dz. (3.24)

Once we reach this point, we can write the solution Φτ0 implicitly for all η > 0 as follows:

Φτ0(η) = Φτ0(0
+)v(η; τ0) +

τ0
h′c

Φ′
τ0(0

+)v′(η; τ0) +
1

h′c

∫ η

0

v′(r; τ0)Q(η − r; τ0) dr, (3.25)

where v(η; τ0) is the solution of the homogeneous equation

τ0v
′′ +Dα

0 [v]− h′cv = 0 with v(0; τ0) = 1, v′(0; τ0) = 0 .

The main properties of this function v(η; τ) are given in Appendix 8 (for any τ). Indeed, by
Lemma 8.1, v(η; τ) is continuous with respect to τ > 0. Thus, we obtain the same continuity with
respect to τ for (3.25) since, in particular, v′(η; τ) and R(η; τ) are continuous at τ0 as well.

Now, as explained above, by continuity with respect to τ around the value τ0, we have that for
all ε > 0 there exists δε > 0 and an interval (−∞, ξετ0) such that, for all τ ∈ (τ0 − δε, τ0 + δε) and
ξ ∈ (−∞, ξετ0), |ϕτ0(ξ) − ϕτ (ξ)| < ε. Now, since ϕτ0(ξ) → ϕc as ξ → ∞, by taking δε smaller if
necessary, we can take ξετ0 large enough so that ϕτ0(ξ) is close to ϕc. Moreover, again, by continuity

and making δε smaller, there exists ξ
ε

τ0 ≫ ξετ0 , such that |ϕτ0(ξ)−ϕc| < ε and |ϕτ0(ξ)−ϕτ (ξ)| < ε

for all τ ∈ (τ0 − δε, τ0 + δε) and ξ ∈ (ξετ0 , ξ
ε

τ0). We can achieve

ξ
ε

τ0 − ξετ0 = O(1/ε) as ε→ 0+, (3.26)

by taking δε smaller if necessary.
By assumption, τ0 ∈ ∂Σc, Σu is an open set and Σ+ is empty. That means, that for all ε > 0,

there exists τ ∈ (τ0 − δε, τ0 + δε) ∩ Σu with

|ϕτ0(ξ)− ϕτ (ξ)| < 2ε for all ξ ∈ (ξετ0 , ξ
ε

τ0). (3.27)

This τ is not unique, it depends on ε (via δε), we skip this in the notation for simplicity, but
we have to bear in mind that by taking other values of ε there is always such a τ .



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 19

By τ̄ ∈ Σu and Lemma 2.8, there exists ξ∗τ̄ , hence η
∗ := ξ∗τ̄ − ξετ0 , such that

Φτ (η) ∼ − C
√
τ

η∗ − η
, as η → (η∗)− . (3.28)

Then Φτ (η) = ϕτ (η + ξετ0)− ϕc, in the interval of existence, can be written as

Φτ (η) = Φτ (0
+)v(η; τ) +

τ

h′c
Φ′

τ (0
+)v′(η; τ) +

1

h′c

∫ η

0

v′(r; τ)Q(η − r; τ) dr , (3.29)

where necessarily η ≤ ξ
ε

τ0 − ξετ0 < η∗ = ξ∗τ̄ − ξετ0 , where ξ
∗
τ̄ depends on τ .

If we take into account this behaviour in (3.28), in terms of η, we notice that the first two terms
in (3.29) are uniformly bounded by a constant proportional to ε: this is because v(η; τ) ∼ η−α and
v′(η; τ) ∼ η−(1+α) as η → ∞ (see Appendix 8 Lemma 8.1), and by |Φτ (0)| < ε and |Φ′

τ (0)| < C,
where this holds by continuity in τ near τ0.

The integral term in (3.29) with R(η; τ) i.e. the linear part of Q(η; τ), can be controlled again
by continuity with respect to τ , since, by construction, this term is close to the one for τ0 (observe
that this term corresponds to the solution with ξ < ξετ0 that is given and close to ϕτ0 in (−∞, ξετ0 ]).
On the other hand, the integral term with the first two terms of Q(η; τ) govern the asymptotic
behaviour for large η but with η < η∗. Let us make this more precise.

Let us assume that η < η∗ and let m > 0 be a small number (that we relate to ε below), and
M ∈ (m, η). Then we can split the integral of (3.29) in the following four terms:∫ η

0

v′(r; τ)Q(η − r; τ)dr =

∫ m

0

v′(r; τ)(Φτ (η − r) + 3ϕc)(Φτ (η − r))2dr (3.30)

+

∫ M

m

v′(r; τ)(Φτ (η − r) + 3ϕc)(Φτ (η − r))2dr (3.31)

+

∫ η

M

v′(r; τ)(Φτ (η − r) + 3ϕc)(Φτ (η − r))2dr (3.32)

+

∫ η

0

v′(r; τ)R(η − r; τ)dr. (3.33)

Takem = εp with p > 1/2, andM = O(1/ε), the latter is possible by (3.26). This last condition
allows to control the term (3.31) by an O(ε) constant: since in this range Φτ satisfies (3.27), the
non-linear term contribution is quadratic. The last non-linear term (3.32) is controlled by the
behaviour of v′(η) for large η (v′(η) ∼ η−(1+α) as η → ∞ and η > M = O(1/ε)) and also by the
fact that in this range Φτ is small and the non-linear contribution is quadratic. As mentioned
above, the last integral (3.33) can be controlled by continuity with respect to τ in v and R, since
this is a perturbation of the similar term in equation (3.25), but Φτ0(η) → 0 as η → ∞, thus it
must be smaller than a constant of O(ε).

Thus, except for the first integral term, which is the right-hand side of (3.30), all other terms
stay below (in absolute value) a constant of order ε (as a consequence, mainly of (3.27) and the
behaviour of v for large η).

Suppose that η∗ = η + Cε where Cε > 0, let us see how the behaviour of such distance to η∗

can be estimated in terms of ε. The right-hand side of (3.30) can be estimated as,∣∣∣ ∫ m

0

v′(r; τ)(Φτ (η − r) + 3ϕc)(Φτ (η − r))2dr
∣∣∣ ≤ Cm2

√
τ

(η∗ − η)3
,

where we have used (3.28) (this is the worst case scenario, where the dominant term of the non-
linear part is (Φτ )

3), and, see Appendix 8 Lemma 8.1, that

v′(r; τ) = O
(h′c
τ
r
)

as r → 0+,

in particular v′(r) < 0 for small, but positive, values of r. So as long as (η∗ − η) > Cε with

Cε ∝ ε
2p−1

3 (τ)1/6, the first term in (3.30) is smaller in absolute value than a constant of order ε.
Then, putting together the estimates on all other terms of (3.29), we obtain |Φτ (η)| < εC.



20 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

Then we can take η larger, thus closer to η∗: Indeed we can get a smaller m, by taking p larger,

to guarantee that |Φτ (η)| < εC by keeping Cε ∝ ε
2p−1

3 (τ)1/6, which is the distance to η∗. Thus
the quadratic behaviour of the non-linear term is valid for as close as we want to η −m, since we
can keep |Φτ (η)| < εC , thus the estimates on the other terms of (3.29) can be done as before for
the new η and m. We can repeat the process for η closer and closer to η∗ by taking a larger p for
any ε.

On the other hand, before blow-up, but close to it, we have that there is a constant C propor-
tional to

√
τ but independent of ε (see Lemma 2.8-(2.18)), such that |ϕτ (ξ)| > C

√
τ/(ξ∗ − ξ) near

ξ∗, or in terms of the variable η

|Φτ (η) + ϕc| > C

√
τ

(η∗ − η)
.

The previous argument allows also |Φτ (η)| < εC near η∗, but then, this would imply that η∗−η >
C
√
τ which contradicts the previous estimates, unless τ (hence τ0) is very small (by making the

balance
√
τ ∼ ε

2p−1
3 (τ)1/6 gives that τ ∼ ε2p−1). If this is the case (taking ε smaller) then τ ∈ Σc

by Lemma 3.6, which gives a contradiction. □

4. Control of the modified problem for small τ : Proof of Theorem 3.5

In this section we prove that solutions of (3.15)-(3.18), provided that τ is sufficiently small,

remain within the range where h and h̃ coincide. As before, we use the notation ϕτ for solutions
to this problem which are constructed as in [1]. Throughout we adopt the notation of Theorem 3.4
and assume that τ is small enough so that the conclusion of this theorem applies.

At this step, we introduce the implicit formulation for the equation (3.15) that is used in the
proofs that follow. We write the equation as the linearised equation around ϕc together with the
remainder terms as we did in the proof of Theorem 1.1. We notice that such formulation for
equation (3.15) is similar, with the obvious changes; the linearised part being the same for h and

h̃. The difference is the choice where we take the shift on ξ that defines the new variable η (for
which we use the same notation) in the current case. As before, we use the notation

h′c := h′(ϕc) = h̃′(ϕc) < 0 .

For the proof of Theorem 3.5 with τ sufficiently small, we shall let ξ0 ∈ R such that ξ0 ≪ ξτ
and, by Theorem 3.4, such that

ϕτ (ξ0)− ϕc > 0 . (4.1)

We can take ξ0 sufficiently away from ξτ , to guarantee that |ϕ′τ (ξ0)| ≪ 1 (by the exponential
behaviour as ξ → −∞) although it is not necessary; it will be enough to have |ϕ′τ (ξ0)| of order
one for τ small. In particular, with such choice we know that ϕ′τ (ξ) < 0 for all ξ ≤ ξ0, but also
for ξ0 < ξ < ξτ .

Let us split the interval of integration of the non-local operator at this ξ0, which gives, as before,

τϕ′′τ +Dα
ξ0 [ϕτ ]− h′cϕτ = h̃(ϕτ )− h′cϕτ − dα

∫ ξ0

−∞

ϕ′τ (y)

(ξ − y)α
dy , (4.2)

where we have used again the notation

Dα
ξ0 [g] := dα

∫ ξ

ξ0

g′(y)

(ξ − y)α
dy.

And the translation of the independent variable is done by means of η = ξ− ξ0 and, as earlier, we
introduce the dependent variable

Φτ (η) = ϕτ (ξ)− ϕc. (4.3)

Applying these changes of variables in (4.2), give the equation:

τΦ′′
τ (η) +Dα

0 [Φτ (η)]− h′cΦτ (η) = Q(η) (4.4)



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 21

where Q is defined, similarly as before, by

Q(η) := h̃(ϕτ (η + ξ0))− h′cΦτ (η)− dα

∫ 0

−∞

Φ′
τ (z)

(η − z)α
dz. (4.5)

Again, we can implicitly write the solution to (4.4)-(4.5), by Appendix 8 and, e.g., [9]. This
gives

Φτ (η) = Φτ (0
+)v(η) +

τ

h′c
Φ′

τ (0
+)v′(η) +

1

h′c

∫ η

0

v′(y)Q(η − y) dy (4.6)

where v is the solution to the homogeneous equation

τv′′ +Dα
0 [v]− h′cv = 0 with v(0) = 1, v′(0) = 0. (4.7)

For simplicity of notation, we shall not write the dependency of v and Q on τ throughout this
section, since essentially τ is fixed.

We recall that the properties and behaviour of v and its derivatives are given in the Appen-
dix 8.1, Lemma 8.2. Observe that v is the same for both linearisations of the associated problems
(1.5) and (3.15) at ϕ = ϕc. We shall need the following lemma.

Lemma 4.1. If Φ′
τ (η) < 0 in the interval (−∞, 0), then for η > 0:∫ 0

−∞

|Φ′
τ (z)|

(η − z)α
dz ≤ C

ηα+1
+
C ′ |Φτ (η)|

ηα
(4.8)

and ∫ 0

−∞

|Φ′
τ (z)|

(η − z)α
dz ≤ C

1 + ηα
. (4.9)

Moreover, for the modified problem (3.15)-(3.18), we have the following upper and lower bounds,
there exists Ch ≥ 0, such that

Q(η) ≥ −Ch

2
Φ2

τ (η) + dα

∫ 0

−M

(−Φ′
τ (z))

(η − z)α
dz (4.10)

for any 0 < M <∞, and

Q(η) ≤ Ch

2
Φ2

τ (η) + dα

∫ 0

−∞

(−Φ′
τ (z))

(η − z)α
dz ≤ Ch

2
Φ2

τ (η)−Dα[Φτ ](0
+). (4.11)

Proof. Observe that (4.8) and (4.9) are obtained as in [13]. The last estimate might be used for
small values of η, and the first one for moderate or large values of η. The constants C in both
estimates are at most of order one, but we cannot guarantee that they are small.

The last two inequalities (4.10) and (4.11), simply follow by applying Taylor’s theorem to h̃(ϕ)

centred at ϕc, since there exists, for each η > 0, ϕ̃η ∈ [infξ∈R ϕτ , ϕ−), such that

h̃(ϕτ (η + ξδ))− h′cΦτ (η) =
h̃′′(ϕ̃η)

2
(Φτ (η))

2. (4.12)

We recall that solutions of (3.15)-(3.18) lie in (ϕ̄, ϕ−), where ϕ̄ < ϕ− is the other zero of (3.17).

Thus infξ∈R ϕτ (ξ) ≥ ϕ̄. Then |h̃′′(ϕ̃η)| ≤ maxϕ∈[ϕ̄,ϕ−] |h̃′′(ϕ)| =: Ch. □

Proof of Theorem 3.5. We recall that limξ→∞ ϕτ (ξ) = ϕc and using the information from Theo-
rem 3.4, we can take ξ0 < ξτ , where ξτ is given in this theorem, such that for τ > 0 sufficiently
small, and since ϕ′ < 0 in (−∞, ξτ ), ϕ(ξ0), ϕ

′(ξ0) are of O(1) if not smaller as τ → 0+, and
certainly we have

ϕτ (ξ0)− ϕc > 0. (4.13)

We shall need to estimate Φ′
τ in (4.3), for which we shall then use the equation

Φ′
τ (η) = Φτ (0

+)v′(η) +
τ

h′c
Φ′

τ (0
+)v′′(η) +

1

h′c

∫ η

0

v′′(y)Q(η − y) dy, (4.14)

that results from differentiating (4.6) and using that v′(0) = 0 by (4.7).



22 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

We will use the results of Lemma 8.2. In particular, we know that v′(η) < 0 for all η if
τ is sufficiently small, that v′′(η) < 0, holds for τ sufficiently small and for η ≤ ηinflex with

ηinflex = O(τ
1

2−α ) and that v′′(η) is non-negative otherwise, see (8.13).

Step 1: Let η := ξ − ξ0, which we consider to be large. In the first step, we need to obtain a
bound for |Φ′

τ (η)ηinflex| for η > ηinflex when τ is sufficiently small. Therefore, we consider (4.14)
and using the estimates of Lemma 8.2, ((8.13) in the integral part, and (8.14)-(8.15) for the other
two terms), and the boundedness of Q, which follows from (4.11). Then, we obtain for η > ηinflex
and τ sufficiently small∣∣Φ′

τ (η)
∣∣ ≤ ∣∣Φτ (0

+)v′(η) +
τ

h′c
Φ′

τ (0
+)v′′(η)

∣∣+ 1

|h′c|

∫ η

0

|v′′(y)||Q(η − y)| dy

≤ K(τ)

ηα+1
+ Cτ +

C ′

|h′c|

(∫ ηinflex

0

(−v′′(y)) dy +
∫ η

ηinflex

v′′(y) dy
)

≲
K(τ)

ηα+1
+ Cτ +

2C ′

|h′c|
(−v′(ηinflex))

≲
ηinflex
τ

∼ τ
α−1
2−α .

Here we are applying the knowledge we have on the sign of v′′ and we are interested in obtaining
a bound of the derivative depending on τ . Notice that this bound, despite having a negative
exponent, is better than the one we obtain from the energy estimate (2.9) which is of order τ−1/2

for τ sufficiently small.
Consequently, we obtain the following estimate, which we will need below, for η > ηinflex,

|Φ′
τ (η)| ηinflex ≲ τ

α
2−α as τ → 0+ . (4.15)

Step 2: In this step we take advantage of the behaviour of v′ near ηinflex for τ sufficiently small.
We prove that v′ behaves as an approximation of the Dirac delta distribution at ηinflex and then
we can approximate

∫ η

0
v′(y)Q(η − y) dy by Q(η − ηinflex) for τ sufficiently small. In fact, in this

step we prove that for τ sufficiently small, there exist an order one constant C > 0 such that∣∣∣ 1
h′c

∫ η

0

v′(y)Q(η − y) dy −Q(η − ηinflex)
∣∣∣ < Cτ

α
2−α . (4.16)

First, we compute the maximum of (−v′), which is attained at the inflection point ηinflex.
Evaluating (−v′) at ηinflex, and using Lemma 8.2 (8.17), we obtain

0 < (−v′(ηinflex)) ∼
|h′c|
τ
ηinflex −

1

(3− α)(2− α)

|h′c|
τ2

η3−α
inflex ∼ τ

α−1
2−α ,

as τ → 0+. Notice that for 0 < α < 1, the previous exponent is negative, implying that this
maximum tends to +∞ as τ → 0+ and the inflection point ηinflex approaches 0.

Next, we derive the rescaling by first translating the original variable η by ηinflex and considering
the behaviour of v′ (8.17) for small values of τ > 0, in particular, for τ ≤ η ≤ ηinflex. Using the
asymptotic behaviour of v′ as τ → 0+, we have for very small values of τ

(−v′(η)) ∼ |h′c|
τ
η − 1

(3− α)(2− α)

|h′c|
τ2

η3−α

∼ |h′c|
τ
ηinflex −

1

(3− α)(2− α)

|h′c|
τ2

η3−α
inflex

+
( |h′c|
τ

− 1

2− α

|h′c|
τ2

η2−α
inflex

)
(η − ηinflex)

∼ C1τ
α−1
2−α + C2

1

τ
(η − ηinflex)

= τ
α−1
2−α

(
C1 + C2

η − ηinflex

τ
1

2−α

)
,

(4.17)

such that η ∈ (ηinflex − ε, ηinflex + ε) for some small ε > 0.



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 23

Let us define, for convenience, the function

ṽ(η − ηinflex) := (−v′(η)),
then we observe that

1 =

∫ ∞

0

(−v′(η)) dη =

∫ ∞

0

ṽ(η − ηinflex) dη. (4.18)

Now we use (4.17) to motivate the following definition (or rescaling of −v′):

(−v′(η)) = 1

τ
1−α
2−α

ω
(η − ηinflex

τ
1

2−α

)
= ωτ (η − ηinflex) ,

such that η ∈ (ηinflex − ε, ηinflex + ε) for some small ε > 0.
To analyse the mollification, we now consider the expression

Q(η) :=
1

h′c

∫ η

0

v′(y)Q(η − y) dy =
1

|h′c|

∫ η

0

(−v′(y))Q(η − y) dy

=
1

|h′c|

∫ ∞

0

ṽ(y − ηinflex)Q(η − y)χ[0,η](y) dy

=

∫ ∞

0

ṽ(y − ηinflex)Qη(η − y) dy,

where Qη(z) :=
1

|h′
c|
Q(z)χ[0,η](η − z). Thus, we integrate at least over (0,∞) to apply (4.18)

Qη(z) =

∫ ∞

0

ṽ(y − ηinflex)Qη(z) dy for z ∈ R. (4.19)

Since Q and, hence, Qη are locally integrable, Lebesgue’s Differentiation Theorem implies that

lim
r→0

1

2r

∫ z+r

z−r

|Qη(y)−Qη(z)| dy = 0

for almost every z ∈ R. In particular, note that the previous identity holds for Q as well.

Now, for a fixed η > 0, τ > 0 small enough and r ≲ 1
2τ

1
2−α , applying the change of variable

z = η − y and (4.19), we obtain

|Q(η)−Qη(η − ηinflex)|

=
∣∣∣ ∫ η

−∞
ṽ(η − (z + ηinflex))(Qη(z)−Qη(η − ηinflex)) dz

∣∣∣
=

∣∣∣ ∫
B(η−ηinflex,r)

ωτ (η − (z + ηinflex))(Qη(z)−Qη(η − ηinflex)) dz

+

∫
B(η−ηinflex,r)c

ṽ(η − (z + ηinflex))(Qη(z)−Qη(η − ηinflex)) dz
∣∣∣ .

On the one hand, by applying the inequality τ
−1
2−α r ≲ 1/2 and the Lebesgue’s Differentiation

Theorem, we obtain the following estimate for a constant C > 0,∣∣∣ ∫
B(η−ηinflex,r)

ωτ (η − (z + ηinflex))(Qη(z)−Qη(η − ηinflex))dz
∣∣∣

≤ 1

τ
1−α
2−α

∫ η−ηinflex+r

η−ηinflex−r

ω
(η − (z + ηinflex)

τ
1

2−α

)
|Qη(z)−Qη(η − ηinflex)|dz

≤ Cτ
α−1
2−α r

( 1

2r

∫
B(η−ηinflex,r)

|Qη(z)−Qη(η − ηinflex)|dz
)

≤ Cτ
α

2−α

2

1

2r

∫
B(η−ηinflex,r)

|Qη(z)−Qη(η − ηinflex)|dz → 0 as r → 0+.

(4.20)

On the other hand, we analyse the complementary part as follows. First, we determine the
complementary of B(η − ηinflex, r) and work on the original variable of integration, we do the
change of variable y = η−z. Then, in the first inequality we apply the estimates on the behaviour



24 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

of v′ (8.14) and (8.17). Finally, we can apply the uniform bound for Q derived from the inequality
(4.11), since Φτ is uniformly bounded in this scenario, and yield the following estimate for some
positive constants C and C ′ holds as r → 0+:∣∣∣ ∫

B(η−ηinflex,r)c
ṽ(η − (z + ηinflex))(Qη(z)−Qη(η − ηinflex))dz

∣∣∣
=

∣∣∣ ∫ η−ηinflex−r

−∞
+

∫ η

η−ηinflex+r

ṽ(η − (z + ηinflex))(Qη(z)−Qη(η − ηinflex))dz
∣∣∣

=
∣∣∣ ∫ ∞

ηinflex+r

+

∫ ηinflex−r

0

ṽ(y − ηinflex)(Qη(η − y)−Qη(η − ηinflex))dy
∣∣∣

≤
∫ ∞

ηinflex+r

K ′(τ)

yα+1
|Qη(η − y)−Qη(η − ηinflex)|dy

+ C̃

∫ ηinflex−r

0

y

τ
|Qη(η − y)−Qη(η − ηinflex)|dy

≤ C1K
′(τ)

(ηinflex + r)α
+
C2(ηinflex − r)2

τ

≤ C
τ

2α
2−α

τ
α

2−α
+ C ′ τ

2
2−α

τ
∼ τ

α
2−α .

(4.21)

Consequently, combining (4.20) and (4.21) gives (4.16).

Step 3: In this step we use the previous steps to estimate (4.6) from below and from above for
η ≳ τ−1/(2−α) for τ sufficiently small.

Recall that η = ξ − ξ0 and note that ξ0 < ξτ is not necessarily large. Consequently, at
η = 0, Φτ and Φ′

τ are not necessarily small. However, by Theorem 3.4, Φτ is decreasing up to an
ητ = ξτ − ξ0 = O(τ−1/(2−α)) for τ > 0 sufficiently small.

Using equation (4.16), we establish the bound

Q(η − ηinflex)− Cτ
α

2−α ≤ 1

h′c

∫ η

0

Q(η − y)v′(y) dy ≤ Q(η − ηinflex) + Cτ
α

2−α (4.22)

for η > ηinflex. Next, we consider the behaviour of v and v′ for small values of τ , (8.14), to yield

f(η) = Φτ (0
+)v(η) +

τ

h′c
Φ′

τ (0
+)v′(η) ∼ Cτ

ηα
(4.23)

for η > ηinflex, where Cτ = O
(
τ

2α
2−α

)
as τ → 0+. Applying the previous estimates (4.22)-(4.23) to

the variation of constants formulation (4.6), we obtain

f(η) +Q(η − ηinflex)− Cτ
α

2−α ≤ Φτ (η) ≤ f(η) +Q(η − ηinflex) + Cτ
α

2−α .

Applying the Mean Value Theorem to Φτ on the interval [η−ηinflex, η], there exists η̃ ∈ (η−ηinflex, η)
such that

f(η) +Q(η − ηinflex)− Cτ
α

2−α ≤ Φτ (η − ηinflex)− Φ′
τ (η̃)ηinflex ≤ f(η) +Q(η − ηinflex) + Cτ

α
2−α .

Using the bound (4.15), we obtain

f(η) +Q(η − ηinflex)− Cτ
α

2−α ≤ Φτ (η − ηinflex) ≤ f(η) +Q(η − ηinflex) + Cτ
α

2−α , (4.24)

for τ sufficiently small.
Before we obtain bounds on the term Q(η − ηinflex), we recall its definition,

Q(η − ηinflex) = h̃ (Φτ (η − ηinflex) + ϕc)− h′cΦτ (η − ηinflex) + dα

∫ 0

−∞

(−Φ′
τ (z))

(η − ηinflex − z)α
dz ,

here the integral term is positive since Φτ is initially decreasing. For large η and a fixed M > 0,
we use Lemma 4.1 to obtain the estimate

0 <
C ′

(η − ηinflex +M)α
≤ dα

∫ 0

−∞

(−Φ′
τ (z))

(η − ηinflex − z)α
dz ≤ C

(η − ηinflex)α
, (4.25)



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 25

where C = (−Φτ (0
+) + ϕ−) and C

′ = (−Φτ (0
+) + Φτ (M)) are positive. These upper and lower

bounds can be combined with the f(η) term but the constants here are not necessarily small,
however, note that we are considering η large.

For the term involving h̃, there are two possibilities depending on its sign. Observe that h̃ is not
convex. In fact, h′′(ϕ) = 0 at ϕ = 0 and ϕc < 0 < ϕ0. If h̃ (Φτ (η − ηinflex) + ϕc)−h′cΦτ (η−ηinflex) ≥
0, then Q remains positive, but, since we know that Φτ (η) → 0 as η → ∞, there exists a larger

η such that h̃ (Φτ (η − ηinflex) + ϕc) − h′cΦτ (η − ηinflex) ≤ 0. In particular, for η − ηinflex ∼ ητ we
obtain to this region: applying the inequality

|ϕτ (η + ξ0)− ϕ0(η + ξ0)| ≲ τ , for η ≲ ητ ,

from Theorem 3.4, we derive that for sufficiently small τ and η large enough (but with η ≲ ητ =
O(τ−1/(2−α))),

|Φτ (η)| ≤ |ϕτ (η + ξ0)− ϕ0(η + ξ0)|+ |ϕ0(η + ξ0)− ϕc| ≲ τ +
1

(η + ξ0)α
.

Thus, for τ sufficiently small, ϕτ can be made arbitrarily close to ϕc with ϕτ decreasing.
For such η’s in a neighbourhood of ητ , we approximate this term by the quadratic expression

h̃ (Φτ (η − ηinflex) + ϕc)− h′cΦτ (η − ηinflex) = h̃′′(ϕ̃)Φ2
τ (η − ηinflex) , (4.26)

then Q can change sign and, in this case, h̃′′(ϕ̃) < 0 (depends on η). Since we know that Φτ is
bounded then

−|h̃′′max| ≤ h̃′′(ϕ̃) ≤ −|h̃′′min| < 0. (4.27)

Subsequently, using the previous estimates (4.25)-(4.27) in (4.24), we derive,

F2(η)− |h̃′′max|Φ2
τ (η − ηinflex)− Cτ

α
2−α ≤ Φτ (η − ηinflex)

≤ F1(η)− |h̃′′min|Φ2
τ (η − ηinflex) + Cτ

α
2−α ,

where

0 <
1

ηα
∼ F2(η) ≤ f(η) + dα

∫ 0

−∞

(−Φ′
τ (z))

(η − ηinflex − z)α
dz ≤ F1(η) ∼

1

ηα
.

Hence, we conclude the following inequalities, rewriting as X = Φτ (η − ηinflex) and considering,
by continuity in η, η ≳ τ−1/(2−α) (but not too away from ητ ) to yield η−α ≲ τα/(2−α),

−C2 τ
α

2−α − C2X
2 ≤ X ≤ C1 τ

α
2−α − C1X

2

for C1, C2, C1 and C2 positive constants. Hence, if we analyse both inequalities and if τ is suffi-
ciently small, so that the term C2τ

α
2−α is small, then both parabolas have real roots and since X

is positive initially, we obtain the inequalities

−1 +
√
1− 4C2τ

α
2−αC2

2C2
≤ Φτ (η − ηinflex) ≤

−1 +
√
1 + 4C1τ

α
2−αC1

2C1
(4.28)

for η in a neighbourhood of ητ . We can extend the argument to all η > ητ by continuity of η,
because (4.28) guarantees that we can extend the previous estimates: notice that the negative
root tends to zero as τ → 0+ and the same happens for the positive root, implying that Φτ (η)
remains close to 0.

Therefore, this allows to control the possible oscillatory behaviour of Φτ and, consequently, of
ϕτ for τ sufficiently small, ensuring that ϕτ remains in the region where h = h̃. □

For completeness we give the following result on the asymptotic behaviour of solutions as η → ∞
for monotone solutions:

Proposition 4.2. If τ > 0 is sufficiently small and ϕτ is decreasing then

lim
ξ→∞

|ϕτ (ξ)− ϕc|ξα < +∞.



26 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

Remark 4.3. In the proof of this proposition we take ξ0, the shift where we split the non-local
term, very large. Then we can derive equation (4.4)-(4.5) similarly and use the implicit formulation
(4.6). We shall use the same notation, but now since we are assuming that ξ0 is very large and
that ϕτ decreases, then the corresponding initial value for Φτ (0

+) is very small.
Now, in this case, by Lemma 4.1 (4.10), as long as Φτ decreases and stays positive (ϕτ > ϕc)

but close to 0 (so h̃(ϕτ ) < 0, but small) then Q is positive,

Q(η) ≥ −Ch

2
(Φτ (0

+))2 + dα
Φτ (−M)− Φτ (0

+)

(η +M)α
. (4.29)

The first term is dominated by the second if we take M ≥ η and large enough, but such that
Mα ≪ (Φτ (0

+))−2, since we also have that 0 < −Φτ (0
+) + Φτ (−M) = −ϕτ (ξ0) + ϕτ (ξ0 −M) ≤

ϕ− − ϕc.

Proof. We fix τ sufficiently small such that Lemma 8.2 of the Appendix 8.1 holds, in particular
0 < v(η) < 1 and v′(η) < 0 for all η > 0.

As said in Remark 4.3, we use the implicit formulation (4.6) (with the same notation as before,
for simplicity). Since here we assume that ϕ′τ < 0 we take ξ0 ≫ 1 (and possibly larger than ξτ of
Theorem 3.4 above). The assumption implies also that Φτ > 0 and that Φ′

τ < 0. Then we follow
an argument similar to that in [13]. Here we can take, if necessary, Φτ (0

+) as small as we want,
by assumption. This means that we can choose the shift ξ0 a posteriori to get the result. For
simplicity of notation we denote

0 < δ := Φτ (0
+).

A lower bound is obtained by applying that Q(η) > 0. Notice that this is possible for very small
δ and large η so that M in (4.29) can be taken 1 ≪ η ≤M and Mα ≪ δ−2. Then:

Φτ (η) ≥ δv(η) +
τΦ′

τ (0
+)

h′c
v′(η)

and this is valid for very large η with η ≪ δ−2/α. The second term is negative, but for all η ≥ η′

such that

η′ ≫ τ

δ

Φ′
τ (0

+)

h′c

then there exists C > 0 such that

Φτ (η) ≥ Cη−α for 1 ≪ η ≪ δ−2/α, (4.30)

with δ ≪ 1 sufficiently small. Here we are using the behaviour of v and its derivatives given in
Lemma 8.2 of the Appendix 8.1. Notice that η′ is not necessarily large and both conditions on η
are compatible, because δ = Φτ (0

+) ≥ C|Φ′
τ (0

+)| and then for τ small enough |Φ′
τ (0

+)|τ/|h′c| ≪
δ2/α−1. Let us obtain an upper bound. Since the second term in (4.6) is negative, we have

Φτ (η) ≤ δv(η) +
1

h′c

∫ η

0

v′(r)Q(η − r) dr .

Then, we can apply the estimates (4.10)-(4.11) of the Lemma 4.1 on Q. But we may split the
integral into several parts. Before that, let us introduce the notation

I1 :=
1

h′c

∫ η

0

v′(r)
(
h̃(ϕτ (η − r + ξ0))− h′cΦτ (η − r)

)
dr,

I2 :=
dα
h′c

∫ η

0

v′(r)

∫ 0

−∞

(−Φ′
τ (z))

(η − r − z)α
dz dr .

We observe that, by hypothesis, we can start for a ξ0 large enough such that h̃′′(ϕ(ξ0)) < 0 so that
I1 ≤ 0 is non-positive. The assumption on ϕτ being decreasing up to ξ0 also implies that I2 ≥ 0.
Then we have that

Φτ (η) ≤ δv(η) + I2. (4.31)



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 27

We obtain an upper bound for I2 using (4.8) and (4.9) of Lemma 4.1 and splitting the interval
of integration at some R > 0:

I2 ≤ dα
h′c

∫ R

0

v′(r)C

(η − r)α+1
+
v′(r)C ′ |Φτ (η − r)|

(η − r)α
dr +

dα
h′c

∫ η

R

Cv′(r)

1 + (η − r)α
dr

≤ C1
R2

2τ

( C

(η −R)α+1
+
C ′ |Φτ (η −R)|

(η −R)α

)
+
C2

α

( 1

Rα
− 1

ηα

)
.

(4.32)

Now, we take R depending on η, once that τ and δ (taken as small as necessary) are fixed:

R(η) = (ση)α/2, for σ ∈ (0, 1).

We take σ such that R(η) ≤ 1. In particular for each η we have

σ < η−1. (4.33)

Then we can say that
1

Rα
≤

( 1

Rα

)2/α

=
1

R2
.

The previous estimate and inequality (4.32) applied to (4.31) yield, for some order one constants,

Φτ (η) ≤ δv(η) +
R2

2τ

C1

(η −R)α+1
+
R2

2τ

C2|Φτ (η −R)|
(η −R)α

+
C3

α

1

R2
− C4

α

1

ηα
. (4.34)

Therefore, we can deduce the following upper bound from (4.34) and using (4.33), where the worst
case scenario is (

1− C2

2τ
σα

)
Φτ (η) ≤

C5

ηα
,

for some C5 > 0 of order one, and, therefore, it is sufficient to take σ small enough such that

σ < τ1/α . (4.35)

Since, we can choose δ = Φτ (0
+) arbitrarily small once τ is fixed and we have (4.30), we can

conclude that for 1 ≪ η ≪ δ−2/α (large enough but in this range), we can take σ satisfying (4.33)
and (4.35), then there exists C > 0 such that

Φτ (η) ≤
C

ηα
for 1 ≪ η ≪ δ−2/α. (4.36)

Finally, taking the limit ξ0 → ∞ implies that δ → 0 so we can increase the range of η in the limit
and we obtain the result. □

5. Numerical computations

In this section we show numerical simulations that confirm the existence of solutions of (1.5)-
(1.7) for a value of τ > 0 under the assumptions (1.11) and (1.14). The numerical experiments are
intended as a complementary tool to illustrate the role of τ as a shooting parameter in the proof
of existence. For definiteness, in this section we consider

ϕ− = 1 , ϕ+ = −0.6 (ϕc = −0.4) , (5.1)

such that all conditions for a non-classical shock wave are satisfied.
First, we show numerical computations of (1.5) performed with the method described and

analysed in [12]. Rewriting the travelling wave problem (1.5) as a system making the change
ψ = ϕ′ gives

ϕ′ = ψ,

τψ′ = h(ϕ)− dα

∫ ξ

−∞

ψ(y)

(ξ − y)α
dy.

(5.2)

The singularity of the integral term Dα[ϕ] is removed by using integration by parts and taking
into account the regularity and far-field behaviour of ϕ, which implies that∫ ξ

−∞

ψ(y)

(ξ − y)α
dy =

1

1− α

∫ ξ

−∞
ψ′(y)(ξ − y)1−α dy. (5.3)



28 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

The initial value problem (5.2)-(5.3) is solved by a scheme using the Heun’s method (see e.g. [8]).
For more information on the numerical scheme see [12].

Next, we proceed by shooting with respect to τ as follows. First, we identify two values of τ , τc
and τu such that τc ∈ Σc and τu ∈ Σu. This is done by integrating the equations for a long enough
interval, typically of length 500, then if the solution approaches the value ϕc = −0.4 in the tail,
we assume that the corresponding τ is in Σc. If the solution decays to negative values beyond say
−10, then we assume that the corresponding τ is in Σu. This operation allows to choose initial
values for τc and τu. Then, we start an iterative process, which consists of computing the solution
for τm = (τu+ τc)/2, and apply the same criteria to either set τm = τc or τm = τu. We repeat this
process as long as |τc − τm| < 10e−15.

Figure 1 shows solutions for α = 0.9, in this case the iteration stops at the value τ ≈ 2.80018.

−20 0 20 40 60 80 100 120 140 160

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

ξ

φ

Figure 1. α = 0.9 and τ ≈ 2.80018

Figure 2 shows solutions for α = 0.5, in this case the iteration stops at the value τ =
72.821821443764975.

−100 0 100 200 300 400

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

ξ

φ

Figure 2. α = 0.5 and τ ≈ 72.82182



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 29

6. Appendix: Continuous dependence on τ

In this section, we prove the continuous dependence on the parameter τ using the general theory
for functional differential equations, see [16, 28, §2]. After rewriting the problem as a functional
differential equation, we check that the necessary hypotheses are fulfilled in order to apply the
auxiliary lemmas and the continuous dependence result from [16, §2].

First, we rewrite (1.5) as a system of first-order delay functional differential equations

ϕ′ = ψ,

ψ′ =
1

τ
h(ϕ)− dα

τ

∫ ξ

−∞

ψ(y)

(ξ − y)α
dy.

To study the continuous dependence of solutions on τ , we add τ as an independent variable.
However, it is easier to consider instead of τ > 0 its inverse ν := 1/τ such that the augmented
system of first-order differential equations reads

ϕ′ = ψ,

ψ′ = νh(ϕ)− ν dα

∫ ξ

−∞

ψ(y)

(ξ − y)α
dy,

ν′ = 0.

(6.1)

To frame this system as one of functional differential equations with finite delay, we split the
integral term at −r for some r > 0:∫ ξ

−∞

ψ(y)

(ξ − y)α
dy =

∫ −r

−∞

ψ(y)

(ξ − y)α
dy +

∫ ξ

−r

ψ(y)

(ξ − y)α
dy.

The first one is treated as a known function of ξ as long as ξ ≥ −r, while the second denotes the
fractional derivative from −r and one needs the initial data to be given on (−r, 0). Without loss
of generality, we consider the function ϕ(ξ− r) to be given for ξ ∈ (−∞, 0) and split the fractional
derivative in two parts as before and applying the change of variable y′ = y + r we obtain

Dα[ϕ] = dα

∫ 0

−∞

ϕ′(y′ − r)

(ξ + r − y′)α
dy′ + dα

∫ ξ+r

0

ϕ′(y′ − r)

(ξ + r − y′)α
dy′

=W (ξ) +Dα
−r[ϕ](ξ),

(6.2)

where W (ξ) := dα
∫ 0

−∞
ϕ′(y′−r)

(ξ+r−y′)α dy
′ is a given function. Using (6.2), system (6.1) can be written

as
ϕ′ = ψ,

ψ′ = νh(ϕ)− ν dα

∫ ξ+r

0

ψ(y′ − r)

(ξ + r − y′)α
dy′ − νW (ξ),

ν′ = 0,

(6.3)

for all ξ ≥ −r. The first and the third equations are ordinary differential equations, whereas the
second one is an integro-differential equation with finite delay in the integral term. Following the
notation of [16, §2], equation (6.3) is a functional differential equation of the form

ẋ = F (ξ, xξ) (6.4)

such that xξ(θ) = x(ξ + θ), for −r ≤ θ ≤ 0.
In our particular case, x = (ϕ, ψ, ν), and F = (F1, F2, F3) is identified as

F1 (ξ, (ϕ, ψ, ν)) = ψ, (6.5)

F2 (ξ, (ϕ, ψ, ν)) = νh(ϕ)− ν dα

∫ ξ+r

0

ψ(y′ − r)

(ξ + r − y′)α
dy′ − νW (ξ), (6.6)

F3 (ξ, (ϕ, ψ, ν)) = 0. (6.7)

Moreover, we consider the operator F as F : D → R3 with domain D ⊆ R × C([−r, 0],R3).
We recall C([−r, 0],R3) is a Banach space with norm ∥φ∥∞ = sup−r≤s≤0 |φ(s)| for functions



30 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

φ ∈ C([−r, 0],R3). Finally, we consider the delay functional differential equation (6.3) for a
starting time σ = 0. As it is mentioned previously, we only need the vector (ϕ, ψ, ν)(θ) for all
−r ≤ θ ≤ 0 as initial datum, the history of ψ(ξ) for ξ < −r is incorporated into W (ξ). Besides,
notice that the application of [1, Lemma 3] gives us the integrability of ψ on (−∞, ξ0) for ξ0 < ξexist
defined in Lemma 2.4. Moreover, this argument proves the finiteness of W (ξ) for all ξ > 0.

We shall show below that we can apply the following theorem to (6.3), rewritten as in (6.4)
with F given by (6.5)-(6.7).

Theorem 6.1 (Continuous dependence [16, Theorem 2.2]). Suppose Ω ⊆ R × C([−r, 0],R3) is
open, (σ0, γ0) ∈ Ω, F 0 ∈ C(Ω,Rn), and x0 is a solution of the problem (6.4) (F 0) with initial
condition (σ0, γ0) which exists and is unique on [σ0 − r, b]. Let V 0 ⊆ Ω be the compact set defined
by

V 0 = {(ξ, x0ξ) : ξ ∈ [σ0, b]}

and let U0 be a neighbourhood of V 0 on which F 0 is bounded. If (σk, γk, F k), k = 1, 2, . . . satisfies
σk → σ0, γk → γ0 and |F k − F 0|U0 → 0 as k → ∞, then there is a k∗ such that the problem
(6.4) (F k) for k ≥ k∗ is such that each solution xk = xk(σk, γk, F k) with initial condition (σk, γk)
exists on [σk − r, b] and xk → x0 uniformly on [σk − r, b]. Since all xk may not be defined on
[σk − r, b], by xk → x0 uniformly on [σk − r, b], we mean that for any ε > 0, there is a k∗(ε) such
that xk, k ≥ k∗(ε), is defined on [σ0 − r + ε, b], and xk → x0 uniformly on [σ0 − r + ε, b].

Next we proceed to check that our F : R × C([−r, 0],R3) → R3 in (6.5)-(6.7) is continuous in
both variables. This is obvious for F1 and F3, since F1 is the projection of the second component
and F3 is just the zero constant function. In the case of F2, the first term, h(ϕ(ξ)), is continuous
because it is a composition of continuous functions. The integral term is continuous since it maps
Cb(R) into Cb(R) and, finally, the last term, W (ξ) is continuous because of the regularity of ϕ in
ξ ∈ (−∞, 0) and finiteness is obtained as is explained above.

To study the existence of solutions for (6.3) starting at σ = 0, we only need to prescribe the
values for (ϕ, ψ, ν)(ξ) at −r < ξ < 0, since the history of ψ(ξ) for ξ < 0 is incorporated in W (ξ)
which we treat as a given function. To study the continuous dependence of solutions on τ (or
equivalently ν) in a neighbourhood of τ0, we consider the following initial data

σk ≡ 0, ϕk → ϕ0, ψk → ψ0, νk → 1
τ0

as k → ∞.

Note that the delicate point is that changing τ (or ν) influences the profile ϕ(ξ), ψ(ξ) = ϕ′(ξ),
for ξ < 0, hence, also F through its dependence on W (ξ). Therefore, we have to use continuous
dependence of ’local’ solutions with respect to τ , to justify the assumptions on ϕk(ξ), ψk(ξ) and
F k. But this results follows from [1, Lemma 2]. Let us apply this here.

Considering the behaviour of the travelling wave solution and its derivative at −∞ we know
that for all νk = 1/τk > 0

lim
ξ→−∞

ϕk(ξ) = lim
ξ→−∞

ϕτk(ξ) = ϕ− and lim
ξ→−∞

ψk(ξ) = lim
ξ→−∞

ϕ′τk(ξ) = 0.

Therefore, by [1, Lemma 2] and for all fixed k > 0 natural number, there exists some ξk =
log(1/k)/λk such that

|ϕk(ξ)− ϕ−| <
1

k
, |ψk(ξ)| < 1

k
, ∀ξ < ξk.

Since it is known that

lim
ξ→−∞

ϕ0(ξ) = lim
ξ→−∞

ϕτ0(ξ) = ϕ− and lim
ξ→−∞

ψ0(ξ) = lim
ξ→−∞

ϕ′τ0(ξ) = 0,

then by the triangle inequality,

|ϕk(ξ)− ϕ0(ξ)| < 2

k
, |ψk(ξ)− ψ0(ξ)| < 2

k
, ∀ξ < ξk.

Now, for each fixed 1/k > 0, we can apply continuous dependence on τ in the interval [ξk, 0] taking
as initial condition an arbitrary sequence of νk = 1/τk that converges to 1/τ0 as k → ∞, σk = ξk,



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 31

F k = F , ϕk = ϕτk and ψk = ϕ′τk . Therefore, by the continuous dependence result, we have that

for each εk = 1
k > 0 there exist k0 > 0, so that k > k0 implies

|ϕk(ξ)− ϕ0(ξ)| < 1

k
and |ψk(ξ)− ψ0(ξ)| < 1

k
, ∀ξ ∈ [ξk, 0].

Since τk → τ0 as k → ∞, it follows that for δk > 0 small, there exists some k0 > 0 such that for all
k > k0, τk ∈ (τ0 − δk, τ0 + δk). If we define a new subsequence taking the values νk for k > k0 and
rename this subsequence again as {νk}k∈N, therefore, for this new sequence and taking σk ≡ 0 we
conclude that

ϕk(ξ) → ϕ0(ξ), ψk(ξ) → ψ0(ξ), ∀ξ ≤ 0,

which is sufficient to apply Theorem 6.1 of continuous dependence on τ for the system (6.3) in an
arbitrary bounded interval [0, b] on the interval of existence.

7. Appendix: Characteristic equations of the linearised problems

Let us recall some results about the zeros of the functions

τz2 + bzα − a for a , b > 0 , α ∈ (0, 1), (7.1)

τz2 + bzα + a for a , b > 0 , α ∈ (0, 1). (7.2)

We can give the following result.

Lemma 7.1. For α ∈ (0, 1), consider the principal branch of zα (−π < arg(z) < π). Then (7.1)
has exactly one positive real root and two complex conjugate roots with negative real part, and (7.2)
has exactly two complex conjugate roots with negative real part on the principal branch of zα.

The statements about (7.1) and (7.2) are proven in [1] and [9], respectively, using variants of
Rouche’s theorem.

For later use, we give the expansion of the zeros of (7.2) provided that a and b are of order 1
as τ → 0+ (see [1]). Namely, as τ → 0+, we have

z = b
1

α−2 e±iπ 1
α−2

1

τ
1

2−α

− a

2b
1

α−2 e±iπ 1
α−2 + b

α−1
α−2αe±iπ α−1

α−2

1

τ
1−α
2−α

+O
( 1

τ
1−2α
2−α

)
. (7.3)

8. Appendix: The linearized equation

In this appendix we consider the linear inhomogeneous equation

τψ′′ +Dα
0 [ψ] + aψ = Q(η) , with a > 0, (8.1)

(here ′ = d
dη ) with initial conditions

ψ(0+) = C0 , ψ′(0+) = C1 . (8.2)

We recall the derivation of a solution via the Laplace transform, see [9]. Applying the Laplace
transform L to (8.1)–(8.2) yields

L(ψ)(s) = 1

τs2 + sα + a

(
L(Q)(s) + (τs+ sα−1)ψ(0+) + τψ′(0+)

)
, (8.3)

where L(f)(s) =
∫∞
0
e−sηf(η)dη. Using L(f ∗ g)(s) = L(f)(s)L(g)(s), we deduce

ψ = ψ(0+)L−1
( τs+ sα−1

τs2 + sα + a

)
+ τψ′(0+)L−1

( 1

τs2 + sα + a

)
+ L−1

( 1

τs2 + sα + a

)
∗Q .

Define

v(η) := L−1
( τs+ sα−1

τs2 + sα + a

)
(η) and ṽ(s) :=

τs+ sα−1

τs2 + sα + a
. (8.4)

Observing that limη→0+ v(η) = lims→∞ sṽ(s) = 1 and

1

τs2 + sα + a
=

1

a
(1− sṽ(s)) = −1

a

(
sL(v)(s)− v(0+)

)



32 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

implies

L−1
( 1

τs2 + sα + a

)
(η) = −1

a
v′(η) .

Consequently,
lim

η→0+
v′(η) = 0 . (8.5)

Writing the expression for ψ in terms of v reads

ψ(η) = ψ(0+)v(η)− τ

a
ψ′(0+)v′(η)− 1

a

∫ η

0

v′(y)Q(η − y) dy . (8.6)

For a > 0, let us sketch the computation of v(η). We recall that since this is the inverse Laplace
transform of ṽ(s), we have to compute

v(η) =
1

2πi

∫
Br

esη
τs+ sα−1

τs2 + sα + a
ds (8.7)

where Br ⊂ C is a Bromwich contour

Br := {s : Re(s) = σ ≥ 1and Im(s) ∈ (−∞,∞)}. (8.8)

Moreover, we restrict to the principal representation of s, namely, here arg(s) ∈ (−π, π]. Following
the approach in [15, 9] and denoting by s1 and s2 = s1 the zeros of (7.2) with b = 1, which are the
poles of the integrand in (8.7). The contribution to the integral of these poles can be computed
away from the Riemann surface cut (since α ∈ (0, 1)) that is the negative part of the real line.
One can then split the integral as follows

v(η) =
a sin(απ)

π

∫ ∞

0

e−ηrK(r) dr + 2Re
(
es1η

τs1 + sα−1
1

2τs1 + αsα−1
1

)
, (8.9)

where

K(r) = rα−1K̃(r), K̃(r) :=
1

(τr2 + a)2 + 2(τr2 + a)rα cos(απ) + r2α
. (8.10)

The integral term is bounded since K ∈ L1((0,∞)). The asymptotic behaviour of the integral
term for η → ∞ can be studied by a refined Watson’s Lemma in [29, p. 65] and [10, §4]. We

note that the function K̃(r) is not differentiable at r = 0, but we have that for a small ε > 0,
the properties K ∈ L1((0,∞)) and K = o(rα−1+ε) for r → 0 imply that L(K)(η) = o(η−α−ε) for
η → ∞. Using a Puiseux series expansion of K(r) for r → 0, allows to deduce for η → ∞,∫ ∞

0

e−ηrK(r) dr =
Γ(α)

a2
1

ηα
+O(η−2α). (8.11)

Finally, we provide the next result of continuity of v(η) = v(η; τ) with respect to τ > 0 and the
asymptotic behaviour at the origin.

Lemma 8.1. For a > 0 and τ > 0, let v(η; τ) be the function defined by (8.9)-(8.10) for η > 0.
Then v(η; τ) as a function of τ is continuous for τ > 0. We also have that v is uniformly bounded
with respect to η > 0 and have the following behaviour for η → 0,

v(η) ∼ 1− a

2τ
η2 , v′(η) ∼ −a

τ
η , as η → 0 .

Proof. The continuity is obtained by the Dominated Convergence Theorem and taking into consid-
eration that the integrand of (8.7) is continuous for τ > 0 since the denominator has two complex
conjugate roots with negative real part (see Lemma 7.1).

While the asymptotic behaviour at η → 0 is derived applying the Initial Value Theorem and
computing the limits in the Laplace transform variable (see e.g. [7, Chapter 2] for more informa-
tion). For τ > 0, we have

lim
η→0

v(η) = lim
s→+∞

sL(v)(s) = lim
s→+∞

s
τs+ sα−1

τs2 + sα + a
= 1 ,

lim
η→0

v′(η) = lim
s→+∞

s (sL(v)(s)− v(0)) = lim
s→+∞

−a s
τs2 + sα + a

= 0 ,



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 33

and, similarly,

lim
η→0

v′′(η) = lim
s→+∞

−a s2

τs2 + sα + a
= −a

τ
. (8.12)

Therefore, we obtain the desired asymptotic behaviour by the Taylor series expansion formula at
the origin. □

8.1. Monotonicity of v for small values of τ . In this section we study the behaviour of v, v′

and v′′. The main idea here is that one can absorb the non-monotone part of each function into
the monotone part for τ > 0 sufficiently small. From [1, Lemma 13 (iii)], we know that the three
functions are uniformly bounded on [0,∞), the first one by a constant independent of τ and the
other two by a constant dependent of τ which gets unbounded as τ → 0+.

Lemma 8.2. For a > 0 and τ > 0, let v(η) be the function defined by (8.9)-(8.10) for η > 0.
Then for τ > 0 sufficiently small, 0 < v(η) < 1, v′(η) < 0 for all η > 0. Moreover, there exists
some ηinflex ∼ τ1/(2−α) as τ → 0+, such that

v′′(η) < 0 for 0 < η < ηinflex and v′′(η) > 0 for η > ηinflex. (8.13)

Also, there exists some η0 ∼ τ
1

2−α as τ → 0+ such that

v(η) ∼ K(τ)

ηα
, v′(η) ∼ −K

′(τ)

ηα+1
for all η > η0 (8.14)

with K(τ), K ′(τ) ∼ τ
2α

2−α as τ → 0+, and

lim
η→+∞

v′′(η) = 0. (8.15)

Finally, for τ ≪ 1 and η → 0+, valid in a layer of η of order τ
1

2−α , we have

v(η) ∼ 1− a

2τ
η2 +

1

(4− α)(3− α)(2− α)

a

τ2
η4−α as η → 0+ , (8.16)

v′(η) ∼ −a
τ
η +

1

(3− α)(2− α)

a

τ2
η3−α as η → 0+ , (8.17)

v′′(η) ∼ −a
τ
+

1

2− α

a

τ2
η2−α as η → 0+ . (8.18)

Proof. Considering the expression (8.9) of v(η), one can obtain the following upper and lower
bounds for the integral term:∫ ∞

0

e−ηr rα−1

(τr2 + a)2 + 2(τr2 + a)rα cos(απ) + r2α
dr

≤
∫ ∞

0

e−ηr rα−1

(τr2 + a)2 sin2(απ)
dr

≤ 1

a2 sin2(απ)
Γ(α)

1

ηα
.

(8.19)

To obtain the first inequality, we rewrite the denominator as

(τr2 + a)2 + 2(τr2 + a)rα cos(απ) + r2α

=
(
(τr2 + a) cos(απ) + rα

)2
+ (τr2 + a)2 sin2(απ),

while the last one is obtained computing the minimum of the denominator which is attained at
zero and applying the change of variable ηr = r′. On the other hand, taking into account that
the integrand is non-negative and proceeding in the same way, one gets this lower bound, for any
0 ≤ A < B∫ ∞

0

e−ηrrα−1K̃(r)dr ≥
∫ B

A

e−ηr rα−1

(τr2 + a+ rα)2
dr ≥ e−ηB(Bα −Aα)

α(τB2 + a+Bα)2
. (8.20)



34 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

We rewrite the second term in (8.9) as follows

Re
(
es1η

τs1 + sα−1
1

2τs1 + αsα−1
1

)
= epη (C1 cos(qη) + C2 sin(qη)) , (8.21)

thus p = Re(s1) < 0 and q = Im(s1), and

C1 = Re
( τs1 + sα−1

1

2τs1 + αsα−1
1

)
, C2 = − Im

( τs1 + sα−1
1

2τs1 + αsα−1
1

)
.

With this notation, we apply the upper bound (8.19) in (8.9), to obtain (observe that sinαπ > 0)

v(η) ≤ Cr(α)
1

ηα
+ 2epηC(τ) ,

with constants

Cr(α) =
Γ(α)

πa sin(απ)
, C(τ) = |C1|+ |C2| .

Observe that the maximum of the function Cr(α) + 2ηαepηC(τ) is attained at ηmax = −α/p > 0,
thus

v(η) ≤ 1

ηα

(
Cr(α) + 2

(
− α

p

)α
e−αC(τ)

)
. (8.22)

We observe that for τ ≪ 1 the constant C(τ) is of order τ
α

2−α . We deduce this fact by applying
(7.3) of the Appendix 7. Indeed, taking into account that τs21 + sα1 = −a, then, as τ → 0+,

C1 = Re
( −a
−2a− (2− α)sα1

)
= Re

( a

2a+ (2− α)sα1

)
=

2a2 + a(2− α)Re(sα1 )

|2a+ (2− α)sα1 |2
= O(τ

α
2−α ) .

To leading order, the sign of Re(sα1 ) ∼ cos(απ/(α−2))τ−α/(2−α) as τ → 0+ depends on α: positive
when α < 2/3, negative when α > 2/3, zero when α = 2/3. We also have, as τ → 0+,

C2 = − Im
( −a
−2a− (2− α)sα1

)
= − Im

( a

2a+ (2− α)sα1

)
=

a(2− α) Im(sα1 )

|2a+ (2− α)sα1 |2
= O(τ

α
2−α )

which is negative to leading order as τ → 0+, since sin(απ/(α−2)) < 0 (see the expansion of s1 with

b = 1 in (7.3) and that Re(s1s
α−1
1 ) = Re(s1s

α−1
1 ) ∼ τ−α/(2−α)(cos(π/(α − 2)) cos((α − 1)π/(α −

2)) + sin(π/(α− 2)) sin((α− 1)π/(α− 2))) + · · · = τ−α/(2−α) cos(π/(α− 2)− (α− 1)π/(α− 2)) =
−τ−α/(2−α)).

This is a good estimate for sufficiently large η. For small and large values of η we have a
uniform, in τ ∈ [0, 1], upper bound. Indeed, for α ∈ (0, 1/2), cos(απ) > 0 and we obtain∫ ∞

0

e−ηrrα−1K̃(r) dr ≤ 1

sin2(απ)

∫ 1

0

rα−1

(τrα + a)2
dr +

∫ ∞

1

rα−1

a2 + r2α
dr

≤ 1

αa2 sin2(απ)
+

1

α
.

For α ∈ [1/2, 1), cos(απ) ≤ 0 and we obtain∫ ∞

0

e−ηrrα−1K̃(r) dr ≤ 1

sin2(απ)

∫ R

0

rα−1

(τrα + a)2
dr +

∫ ∞

R

rα−1

(τr2 + a− rα)2
dr

≤ Rα

αa2 sin2(απ)
+ sup

r∈(R,∞)

{ r2α

(τr2 + a− rα)2

}R−α

α

where R is larger that the positive root of τr2 + a − rα. Then, for all α ∈ (0, 1) and η > 0 with
τ ≪ 1,

v(η) ≤ C + C(τ) +O(τ) , (8.23)

for an order one constant C, and C(τ) = O(τα/(2−α)). The same bounds, clearly hold replacing
v(η) by |v(η)|. Let us see that, indeed v(η) > 0 for all η if τ is sufficiently small.

We also observe that although p < 0, in the limit τ → 0+ we have the following behaviours:

ηRe(s1) → 0, if η ≪ τ
1

2−α ,



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 35

ηRe(s1) → −C, if η ∼ τ
1

2−α ,

ηRe(s1) → −∞ if η > τ
1

2−α .

This follows from Lemma 7.1 in Appendix 7.
In the third case we then have as a lower bound for v, using (8.20) with A = 0 and B = 1/η

1

ηα

(a sin(απ)
eαπ

η4

(τ + aη2 + η2−α)2
− 2epηηαC(τ)

)
≤ v(η) .

Since the function η4

(τ+aη2+η2−α)2 is increasing and the minimum of the second term is attained

at ηmax = −α/p = O(τ1/(2−α)) we have that there exists η0 > ηmax with limτ→0+ η0/ηmax = ∞,
such that

0 <
1

ηα

(a sin(απ)
eαπ

η40
(τ + aη20 + η2−α

0 )2
− 2epη0ηα0C(τ)

)
≤ v(η) for all η ≥ η0.

We can improve this for η0 = Kηmax with K > 1 sufficiently large, with the estimate

0 <
1

ηα

(a sin(απ)
eαπ

η40
(τ + aη20 + η2−α

0 )2
− 2epηmaxηαmaxC(τ)

)
≤ v(η) for all η ≥ η0

since both terms are of order τ2α/(2−α) in that case, but the first has the freedom for K which
can be made large, but independent of τ . This is because

η40
(τ + aη20 + η2−α

0 )2
∼ τ2α/(2−α) K4

(1 + aα2C2K2τα/(2−α) +K2−αα2−αC2−α)2
, τ → 0+ .

This, (8.22) and (8.23) imply the first behaviour given in (8.14).
More generally, from (8.20) with A = 0, taking 0 < B < |p|, we have

e−Bη
(a sin(απ)

π

Bα

α(τB2 + a+Bα)2
− 2eη(−|p|+B)C(τ)

)
≤ v(η),

so, for all η ∈ (0, η0),

e−Bη
(a sin(απ)

π

Bα

α(τB2 + a+Bα)2
− 2C(τ)

)
≤ v(η);

then we can choose B = O(1) as τ → 0+ to guarantee that the right-hand side is positive. This
shows that v(η) > 0 for all η > 0 if τ is small enough.

Let us now get the behaviour and bounds on v′(η). Using (8.9) and (8.21) we derive

v′(η) = −a sin(απ)
π

∫ ∞

0

e−rηrαK̃(r)dr

+ 2epη ((p cos(qη)− q sin(qη))C1 + (p sin(qη) + q cos(qη)C2)

= −a sin(απ)
π

∫ ∞

0

e−rηrαK̃(r)dr +Re
(
eiqηs1

τs1 + sα−1
1

2τs1 + αsα−1
1

)
epη

= −a sin(απ)
π

∫ ∞

0

e−rηrαK̃(r)dr − Re
(
eiqη

a

2τs1 + αsα−1
1

)
epη .

Then, we have the lower bounds for v′ (that are obtained in a similar way as the upper bounds
on v (8.22) and (8.23)):

v′(η) ≥ − 1

ηα+1

( Γ(α+ 1)

aπ sin(απ)
+
(α+ 1

|p|

)α+1

e−(α+1)C ′(τ)
)
, (8.24)

v′(η) ≥ − τ−1

2(1− α) sin(απ)
− 1

(α+ 1)πa sin(απ)
− 2C ′(τ)epη , (8.25)

where C ′(τ) = 2(|C1|+ |C2|)(|p|+ |q|) = O(τ−
1−α
2−α ) as τ → 0+.



36 F. ACHLEITNER, C. M. CUESTA, X. DIEZ-IZAGIRRE EJDE-2025/45

Then, we use that for any 0 ≤ A < B,∫ ∞

0

e−ηrrαK̃(r)dr ≥
∫ B

A

e−ηrrα

(τr2 + a+ rα)2
dr ≥ e−ηB(Bα+1 −Aα+1)

(α+ 1)(τB2 + a+Bα)2
. (8.26)

Taking A = 0 and B = 1/η, we obtain that there exists η′0 > η′max = −(α+1)/p such that, for all
η ≥ η′0,

v′(η) ≤ − 1

ηα+1

( a sin(απ)
e(α+ 1)π

(η′0)
4

(τ + a(η′0)
2 + (η′0)

2−α)2
− 2epη

′
0(η′0)

α+1C ′(τ)
)
< 0 .

Which can again be improved, as before, for τ sufficiently small, for η′0 = K ′η′max for some K ′ > 1
sufficiently large, but independent of τ , so that for all η ≥ η′0,

v′(η) ≤ − 1

ηα+1

( a sin(απ)
e(α+ 1)π

(η′0)
4

(τ + a(η′0)
2 + (η′0)

2−α)2
− 2epη

′
max(η′max)

α+1C ′(τ)
)
< 0.

In this case the term in brackets is also of the order τ
2α

2−α as τ → 0+. This implies the second
estimate in (8.14).

The behaviour of v′′(η) for η > η0 can be deduced similarly, but it is enough for our purposes
to just get (8.15). This can be done by using similar estimates as in (8.19) and (8.21).

Let us now show that v′(η) < 0 for η ≤ η0. For very small values of η, we can take B = K|p|
and choose K > 1 large enough, but independent of τ , so that for τ sufficiently small, we obtain
that ( a sin(απ)

e(α+ 1)π

Bα+1

(τB2 + a+Bα)2
− 2C ′(τ)

)
> 0

observe that both terms are of the same order as τ → 0+, but making K large makes the first
larger. Thus, there exists K > 1 large enough so that for all 0 < η ≤ (K|p|)−1 and for τ small
enough, we have v′(η) ≤ 0, which together with the previous estimate implies that v decreases for
all η > 0.

A next order correction of the behaviours given in Lemma 8.1 and of that implied by (8.12),
can be obtained from a similar computation, as in the proof of this lemma, of the third derivative.
Putting all together we obtain the expansion (8.18) for v′′. Thus there is a value ηinflex ≪ 1 if τ
is sufficiently small such that v′′(ηinflex) = 0 and has

ηinflex ∼ (2− α)
1

2−α τ
1

2−α as τ → 0+. (8.27)

Then, from this and (8.18) we deduce (8.17), and then (8.16). These limits are valid as long as
η ≤ ηinflex for τ small enough but positive.

From the linear equation satisfied by v, which is τv′′ + Dα
0 [v] + av = 0 with v(η) > 0 and

v′(η) < 0 for all η > 0 if τ is sufficiently small, we deduce that τv′′ + av > 0 for all η > 0 if τ is
sufficiently small. If initially v′′ < 0 and on the other hand v > 0 decreases for all η, then v′′ must

change sign. The regularity of v and the fact that ηmax and η′max are of order τ
1

2−α as τ → 0+,
imply that for τ small enough this change of sign of v′′ occurs only once, and that might be given
by (8.27).

We observe that there is a boundary layer of size O(τ1/(2−α)) as τ → 0+. In particular the
behaviours obtained above are consistent for η ∼ ηinflex with the behaviour of the corresponding
solution v0 of the linear problem when τ = 0 as η → 0, since (see [15])

lim
η→0

v0(η) = 1 and v′0(η) ∼ −aηα−1 as η → 0+. □

Acknowledgments. F. Achleitner was supported by the Austrian Science Fund (FWF) via the
FWF-funded SFB 10.55776/F65. C. M. Cuesta and X. Diez-Izagirre were supported by grant
PID2021-126813NB-I00 from MICIU/AEI/10.13039/501100011033, by “ERDF A way of making
Europe”, and by the Basque Government through the Research Group IT1615-22. X. Diez-Izagirre
was also supported by the Basque Government through the pre-doctoral grant PRE-2018-2-0013.



EJDE-2025/45 NON-LOCAL KDV-BURGERS UNDERCOMPRESSIVE WAVES 37

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Franz Achleitner
Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Haupt-

strasse 8-10, 1040 Wien, Austria
Email address: franz.achleitner@tuwien.ac.at

Carlota M. Cuesta

University of the Basque Country (UPV/EHU), Faculty of Science and Technology, Department of
Mathematics, Aptdo. 644, 48080 Bilbao, Spain

Email address: carlotamaria.cuesta@ehu.eus

Xuban Diez-Izagirre

University of the Basque Country (UPV/EHU), Faculty of Economics and Business, Department of

Applied Economics, Oñati plaza 1, 20018 Donostia-San Sebastian, Spain
Email address: xuban.diez@ehu.eus


	1. Introduction
	2. Preliminary results
	2.1. The non-local operator and some elementary lemmas
	2.2. Existence of trajectories that satisfy (1.6) and the derivation of (1.14)

	3. Existence of undercompressive waves: Proof of Theorem 1.1
	3.1. The set u
	3.2. The sets c and +: end of the proof

	4. Control of the modified problem for small : Proof of Theorem 3.5
	5. Numerical computations
	6. Appendix: Continuous dependence on 
	7. Appendix: Characteristic equations of the linearised problems
	8. Appendix: The linearized equation
	8.1. Monotonicity of v for small values of 
	Acknowledgments

	References