Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 97, pp. 1–26.

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

DOI: 10.58997/ejde.2025.97

INVERSE SCATTERING METHOD FOR AN INTEGRABLE SYSTEM OF

DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS

MEHMET UNLU

Abstract. We present a method for solving the integrable system of nonlinear partial differen-

tial equations, known as the derivative nonlinear Schrödinger II system (DNSL II system), also
called the Chen-Lee-Liu system. This is done by presenting a solution technique for the inverse

scattering problem for the corresponding linear system of ordinary differential equations with

energy-dependent potentials. The relevant inverse scattering problem is solved by establishing a
system of linear integral equations, which we refer to as the Marchenko system of linear integral

equations. In solving the inverse scattering problem we use the input data set consisting of a
transmission coefficient, a reflection coefficient, and the bound-state information presented in

the form of a pair of matrix triplets. Using our data set as input to the Marchenko system, we

recover the potentials from the solution to the Marchenko system. By using the time-evolved
input data set, we recover the time-evolved potentials, where those potentials form a solution

to the integrable DNLS II system.

1. Introduction

In this article we present a technique for solving the nonlinear system of partial differential
equations

iqt + qxx − iqqxr = 0,

irt − rxx − iqrrx = 0,
(1.1)

where x and t are the independent variables taking values on the real axis R, the subscripts
denote the respective partial derivatives, the dependent variables q and r are complex-valued
functions of x and t. We refer to x as the spacial coordinate and t as the time coordinate.
The nonlinear system (1.1) is known as the derivative nonlinear Schrödinger II system (DNLS II
system) [1, 3, 4, 12, 13, 17, 23, 30, 31, 32, 33, 34]. It is also known as the Chen-Lee-Liu system. It
has important physical applications in propagation of electromagnetic waves in nonlinear media,
propagation of hydromagnetic waves traveling in a magnetic field, and transmission of ultra short
nonlinear pulses in optical fibers [1, 22].

The nonlinear system (1.1) is related to the nonlinear system

iq̃t + q̃xx − i(q̃2r̃)x = 0,

ir̃t − r̃xx − i(q̃r̃2)x = 0,
(1.2)

which is known as the DNSL I system [1, 4, 16, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29] or the Kaup-
Newell system. We use a tilde to denote the quantities related to (1.2). We observe that the linear
parts in (1.1) and (1.2) coincide but their nonlinear parts differ from each other. Both nonlinear
systems (1.1) and (1.2) are known to be integrable in the sense of the inverse scattering transform
[6, 21, 25]. The integrability of (1.1) is assured [1, 2, 4, 29] by the existence of the corresponding
AKNS pair (X , T ), where X and T are the 2 × 2 matrix-valued functions containing q and r,
their partial x-derivatives, and the spectral parameter denoted by ζ. Similarly, the integrability

2020 Mathematics Subject Classification. 35Q55, 37K10, 37K15, 37K30, 34A55, 34L25, 34L40, 47A40.
Key words and phrases. Inverse scattering; first-order linear system; Marchenko method;
derivative nonlinear Schrödinger equations; Chen-Lee-Liu system.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted July 23, 2025. Published October 14, 2025.

1



2 M. UNLU EJDE-2025/97

of (1.2) is assured by the existence of the corresponding AKNS pair (X̃ , T̃ ), where X̃ and X̃ are
the 2× 2 matrix-valued functions containing q̃ and r̃, their partial x-derivatives, and the spectral
parameter ζ.

The matrix X that appears in the first-order linear system of the differential equations given
by

d

dx

[
α
β

]
= X

[
α
β

]
, x ∈ R, (1.3)

where the quantities α and β are the two components of the wave function depending on the
independent variable x. The independent variable t in (1.1) appears in (1.3) as a parameter. The
AKNS pair matrices X and T associated with (1.1) are

X =

[
−iζ2 ζq
ζr iζ2 + i

2qr

]
, (1.4)

T =

[
−2iζ4 − iζ2qr 2ζ3q + ζ

(
iqx + 1

2q
2r
)

2ζ3r + ζ
(
−irx + 1

2qr
2
)

2iζ4 + iζ2qr + 1
2 (qrx − qxr) +

i
4q

2r2

]
. (1.5)

Using (1.4) in (1.3), we write the corresponding first-order linear system as

d

dx

[
α
β

]
=

[
−iζ2 ζq
ζr iζ2 + i

2qr

] [
α
β

]
, x ∈ R. (1.6)

The matrices X and T appearing in (1.4) and (1.5) form the AKNS pair for the nonlinear system
(1.1) in the sense that the 2× 2 matrix-valued system of equations

Xt − Tx + XT − T X = 0, (1.7)

yields (1.1). In other words, by using the matrices X and T , we evaluate the 2×2 matrix appearing
on the left-hand side of (1.7), where each entry is a polynomial in ζ. Those polynomials identically
vanish for all x, t, and ζ provided that q and r satisfy the nonlinear system (1.1).

In a similar manner, the 2 × 2 AKNS pair matrices X̃ and T̃ associated with the nonlinear
system (1.2) are

X̃ =

[
−iζ2 ζq̃
ζr̃ iζ2

]
, T̃ =

[
−2iζ4 − iζ2q̃r̃ 2ζ3q̃ + ζ(iq̃x + q̃2r̃)

2ζ3r̃ + ζ(−ir̃x + q̃r̃2) 2iζ4 + iζ2q̃r̃

]
.

The matrix X̃ appears in the first-order linear system of differential equations given by

d

dx

[
α̃

β̃

]
= X̃

[
α̃

β̃

]
, (1.8)

where x is the independent variable, t appears as a parameter, and the dependent variables α̃
and β̃ are functions x and they also contain the parameter t. We write the linear system (1.8)
explicitly as

d

dx

[
α̃

β̃

]
=

[
−iζ2 ζq̃
ζr̃ iζ2

] [
α̃

β̃

]
, x ∈ R. (1.9)

Because of the appearance of the spectral parameter ζ in X and X̃ , the linear systems (1.6) and
(1.9) also depend on the spectral parameter ζ. We refer to q and r appearing in (1.6) as the
potentials. Since q and r appear in (1.6) in the form of ζq and ζr, we refer to the potentials in
(1.6) as energy-dependent potentials. This is because the spectral parameter ζ appearing in (1.6)
has the physical interpretation as energy.

In this article we relate the solutions to the linear systems (1.6) and (1.9) through the trans-
formation [

α
β

]
= c

[
1 0
0 E(x)

] [
α̃

β̃

]
, (1.10)

where c is an arbitrary constant and E(x) is the complex-valued scalar quantity given as

E(x) := exp
( i
2

∫ x

−∞
dy q(y) r(y)

)
. (1.11)



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 3

The arbitrary constant c is determined by a spacial asymptotic condition when we use a particular
solution. Since the quantity E(x) defined in (1.11) is in general complex valued, it does not
necessarily have the unit amplitude. From (1.11) it follows that

lim
x→−∞

E(x) = 1, lim
x→+∞

E(x) = eiµ/2, (1.12)

where we have defined the complex constant µ as

µ :=

∫ ∞

−∞
dy q(y) r(y). (1.13)

Our goal is to analyze the direct and inverse scattering problems for the linear system (1.6) with
the goal of developing a solution method for the nonlinear system (1.1). To present our method
in the simplest way, we assume that the potentials q and r in (1.6) belong to the Schwartz class
in x ∈ R for each fixed t. We recall that the Schwartz class S(R) consists of functions of x where
the derivatives of all orders exist and are continuous and those derivatives vanish as x → ±∞
faster than any negative power of |x|. Although our results hold under weaker conditions on the
potentials, we present our ideas in the simplest form by assuming that q and r belong to S(R).
Since t appears as a parameter in the analysis of the direct and inverse scattering problems for
(1.6), we mostly suppress the t-dependence for the related quantities. For example, we write q(x)
and r(x) instead of q(x, t) and r(x, t) in Sections 1-4. On the other hand, we display the argument
t in Sections 5 and 6 for further clarity.

When the potentials q and r belong to the Schwartz class S(R), we have a scattering scenario
associated with (1.6). In other words, the potentials q and r vanish as x → ±∞, and hence

any solution to (1.6) for ζ ∈ R behave as a linear combination of the column vectors

[
e−iζ2x

0

]
and

[
0

eiζ
2x

]
. As indicated in Section 2, this allows us to use some particular solutions to (1.6)

known as the Jost solutions. From the spacial asymptotics of the Jost solutions, we introduce
the scattering coefficients that are functions of the spectral parameter ζ. For ζ ∈ R, we know
that (1.6) has only two linearly independent column-vector solutions. The linear system (1.6) may
also have column-vector solutions that are square integrable in x ∈ R. Such solutions are known
as bound-state solutions to (1.6), and they occur only at certain complex values of the spectral
parameter ζ. At each bound-state ζ-value, it is possible to have two or more linearly independent
solutions to (1.6). The number of such linearly independent solutions determines the multiplicity
of the bound state at that ζ-value. We refer to the bound-state ζ-values, their multiplicities, and
the normalization constants associated with each of the bound-state multiplicity collectively as
the bound-state information.

The direct scattering problem (1.6) consists of the determination of the scattering coefficients
and the bound-state information when the potentials q and r are known. On the other hand,
the inverse scattering for (1.6) consists of the determination of the potentials q and r when the
scattering coefficients and the bound-state information are known. When the scattering coefficients
and the bound-state information contain the parameter t compatible with the AKNS pair matrix
T appearing in (1.5), the solution to the inverse problem for (1.6) yields the potentials q and r
that also contain the parameter t. In that case, those time-evolved potentials q and r satisfy the
nonlinear partial differential equations given in (1.1).

This article is organized as follows. In Section 2 we present the four Jost solutions to (1.6) and
briefly describe their relevant properties. Using the relationship (1.10), we relate the Jost solutions
to (1.6) to the Jost solutions to the linear system given in (1.9). As indicated in Theorem 2.1, we
determine that the potential pair (q, r) appearing in (1.6) is related to the potential pair (q̃, r̃) as in
(2.8). This helps us to determine the properties of the Jost solutions and the scattering coefficients
for (1.6) in terms of the known properties [12, 13] of the corresponding quantities associated with
(1.9). In Section 3 we present the relevant properties of the bound-state information for (1.6)
by exploiting the connection between (1.6) and (1.9) established in Section 2. In Section 4 we
introduce a Riemann–Hilbert problem related to the inverse scattering problem for (1.6). This is



4 M. UNLU EJDE-2025/97

done by relating a pair of Jost solutions to (1.6) to another pair of Jost solutions, where the func-
tional relationship involves the scattering coefficients for (1.6). With the help of the appropriate
properties of the Jost solutions and using a Fourier transformation, we derive a Marchenko system
[7, 8, 9, 10, 11, 14, 15] of linear integral equations associated with (1.6). We then show how the
potentials q and r in (1.6) are constructed from the solution to the Marchenko system with the
input data set consisting of the scattering coefficients and the bound-state information for (1.6).
In fact, we show that it is sufficient to use the right scattering coefficients in our input data set
as the left scattering coefficients are uniquely determined by the right scattering coefficients. In
Section 5 we obtain solutions to the nonlinear system (1.1) with the help of the solution to the
Marchenko system with the time-evolved input data set. Finally, in Section 6 we illustrate the
solution method of Section 5 with two explicit examples.

2. The direct scattering problem

In this section we study the direct scattering problem for the linear system (1.6) associated with
the Chen-Lee-Liu system (1.1). We recall that for each fixed t ∈ R, the potentials q and r in (1.6)
belong to the Schwartz class S(R) in x ∈ R. As already indicated, we suppress the dependence on
the parameter t in the quantities related to (1.6) and (1.9).

We first introduce the four Jost solutions, which are the particular column-vector solutions to
(1.6), denoted by ψ(ζ, x), ψ̄(ζ, x), ϕ(ζ, x), ϕ̄(ζ, x), respectively. We use the subscripts 1 and 2 for
the first and second components of the Jost solutions, respectively, i.e. we let

ψ(ζ, x) =

[
ψ1(ζ, x)
ψ2(ζ, x)

]
, ψ̄(ζ, x) =

[
ψ̄1(ζ, x)
ψ̄2(ζ, x)

]
, (2.1)

ϕ(ζ, x) =

[
ϕ1(ζ, x)
ϕ2(ζ, x)

]
, ϕ̄(ζ, x) =

[
ϕ̄1(ζ, x)
ϕ̄2(ζ, x)

]
. (2.2)

The Jost solutions to (1.6) are those solutions that satisfy the respective spacial asymptotics[
ψ1(ζ, x)
ψ2(ζ, x)

]
=

[
o(1)

eiζ
2x
[
1 + o(1)

]]
, x→ +∞, (2.3)

[
ψ̄1(ζ, x)
ψ̄2(ζ, x)

]
=

[
e−iζ2x

[
1 + o(1)

]
o(1)

]
, x→ +∞, (2.4)

[
ϕ1(ζ, x)
ϕ2(ζ, x)

]
=

[
e−iζ2x

[
1 + o(1)

]
o(1)

]
, x→ −∞, (2.5)

[
ϕ̄1(ζ, x)
ϕ̄2(ζ, x)

]
=

[
o(1)

eiζ
2x
[
1 + o(1)

]]
, x→ −∞. (2.6)

We remark that in our notation, the bar over a quantity does not denote complex conjugation.
We introduce the auxiliary spectral parameter λ in terms of the spectral parameter ζ as

λ = ζ2, ζ =
√
λ, (2.7)

where the square root denotes the principal part of the complex-valued square root function. We
use C to denote the complex plane, C+ for the upper-half complex plane, C− for the lower-half
complex plane, and we let C+ := C+ ∪ R and C− := C− ∪ R.

In the next theorem, we describe the relevant properties of the Jost solutions to (1.6) concerning
their existence and their domains of analyticity and continuity in ζ. Since we analyze those
properties for each fixed t ∈ R, we do not separately mention that the properties hold for each
fixed t ∈ R.

Theorem 2.1. Assume that the potentials q and r in (1.6) belong to the Schwartz class S(R) in
x ∈ R. Let the spectral parameter ζ be related to the parameter λ as in (2.7). Then, we have the
following:



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 5

(a) The Jost solutions ψ(ζ, x), and ϕ(ζ, x) to (1.6) exist, are analytic in the first and third
quadrants in the complex ζ-plane, and are continuous in the closures of those quadrants for each
fixed x ∈ R. Similarly, the Jost solutions ψ̄(ζ, x) and ϕ̄(ζ, x) to (1.6) exist, are analytic in the
second and fourth quadrants in the complex ζ-plane, and are continuous in the closures of those
quadrants for each fixed x ∈ R.

(b) The four Jost solution components ψ1(ζ, x), ψ̄2(ζ, x), ϕ̄1(ζ, x), and ϕ2(ζ, x) are odd in ζ
whereas the other four Jost solution components ψ̄1(ζ, x), ψ2(ζ, x), ϕ1(ζ, x), and ϕ̄2(ζ, x) are even
in ζ. Furthermore, for each fixed x ∈ R, the four scalar functions ψ1(ζ, x)/ζ, ψ2(ζ, x), ϕ1(ζ, x),
ϕ2(ζ, x)/ζ are even in ζ, and hence they are functions of λ. As functions of λ, those four scalar

quantities are analytic in λ ∈ C+ and continuous in λ ∈ C+. Similarly, for each fixed x ∈ R,
the four scalar functions ψ̄1(ζ, x), ψ̄2(ζ, x)/ζ, ϕ̄1(ζ, x)/ζ, ϕ̄2(ζ, x) are even in ζ, and hence they
are functions of λ. As functions of λ, those four scalar quantities are analytic in λ ∈ C− and
continuous in λ ∈ C−.

Proof. The proof is obtained by proceeding as in [12, Theorem 2.2], where the corresponding
results are obtained for the linear system (1.9). □

In the next theorem, we present the relationship between the potential pair (q, r) in (1.6) and
the potential pair (q̃, r̃) in (1.9), when the solutions to (1.6) and to (1.9) are related to each other
as in (1.10).

Theorem 2.2. Assume that the potentials q and r appearing in (1.6) belong to the Schwartz class
S(R) in x ∈ R. Further assume that solutions to (1.9) are related to the solutions to (1.6) as in
(1.10). Then, the potential pair (q, r) is related to the potential pair (q̃, r̃) as

q(x) = E(x)−1q̃(x), r(x) = E(x) r̃(x), (2.8)

where E(x) is the quantity defined in (1.11). Consequently, the potential pair (q̃, r̃) also belongs
to the Schwartz class S(R).

Proof. Taking the x-derivative on both sides in (1.10), we obtain[
α
β

]′
= c

[
0 0
0 E′(x)

] [
α̃

β̃

]
+ c

[
1 0
0 E(x)

] [
α̃

β̃

]′
, (2.9)

where we recall that c is a constant independent of x but may depend on the spectral parameter
ζ. Using (1.6) on the left-hand side of (2.9) and using (1.9) on the right-hand side, we obtain[

−iζ2 ζq(x)
ζr(x) iζ2 + i

2q(x)r(x)

] [
α
β

]
= c

[
0 0
0 E′(x)

] [
α̃

β̃

]
+ c

[
1 0
0 E(x)

] [
−iζ2 ζq̃(x)
ζr̃(x) iζ2

] [
α̃

β̃

]
. (2.10)

Next, using (1.10) on the left-hand side of (2.10), we obtain

c

[
−iζ2 ζq(x)
ζr(x) iζ2 + i

2q(x)r(x)

] [
1 0
0 E(x)

] [
α̃

β̃

]
= c

[
0 0
0 E′(x)

] [
α̃

β̃

]
+ c

[
−iζ2 ζq̃(x)

ζr̃(x)E(x) iζ2E(x)

] [
α̃

β̃

]
.

(2.11)

Since c

[
α̃

β̃

]
is an arbitrary solution to (1.9), from (2.11) we obtain[

−iζ2 ζq(x)E(x)
ζr(x) iζ2E(x) + i

2 q(x)r(x)E(x)

]
=

[
−iζ2 ζq̃(x)

ζr̃(x)E(x) E′(x) + iζ2E(x)

]
. (2.12)

By taking the x-derivative of both sides of (1.11) we have

E′(x) =
i

2
q(x)r(x)E(x). (2.13)

Comparing the individual entries of the matrix equality in (2.12), with the help of (2.13), we
observe that (2.8) holds. With the help of (1.11) we confirm that the potentials q̃ and r̃ belong
to the Schwartz class S(R) when the potentials q and r belong to S(R). □



6 M. UNLU EJDE-2025/97

As indicated in Theorem 2.2, the potential pair (q̃, r̃) appearing in (1.9) belongs to the Schwartz
class S(R) when the potential pair (q, r) appearing in (1.6) belongs to S(R). We introduce the
Jost solutions to (1.9) in the same manner the Jost solutions to (1.6) are introduced. For the

Jost solutions associated with (1.9), we use the notation ψ̃(ζ, x), ˜̄ψ(ζ, x), ϕ̃(ζ, x), and ˜̄ϕ(ζ, x) by
requiring that those solutions satisfy the respective spacial asymptotics[

ψ̃1(ζ, x)

ψ̃2(ζ, x)

]
=

[
o(1)

eiζ
2x
[
1 + o(1)

]]
, x→ +∞, (2.14)

[
˜̄ψ1(ζ, x)
˜̄ψ2(ζ, x)

]
=

[
e−iζ2x

[
1 + o(1)

]
o(1)

]
, x→ +∞, (2.15)

[
ϕ̃1(ζ, x)

ϕ̃2(ζ, x)

]
=

[
e−iζ2x

[
1 + o(1)

]
o(1)

]
, x→ −∞, (2.16)[

˜̄ϕ1(ζ, x)
˜̄ϕ2(ζ, x)

]
=

[
o(1)

eiζ
2x
[
1 + o(1)

]]
, x→ −∞, (2.17)

which are the analogs of (2.3)–(2.6), respectively. We refer the reader to [12, 13] where it is shown
that the Jost solutions to (1.9) have the same properties listed in Theorem 2.1 and satisfied by
the Jost solutions to (1.6).

In the following theorem, we show the connection between the Jost solutions to (1.6) and the
Jost solutions to (1.9)

Theorem 2.3. Let the potential pair (q, r) in (1.6) and the potential pair (q̃, r̃) in (1.9) are related
to each other as in (2.8), where E(x) and µ are the quantities appearing in (1.11) and (1.13),
respectively. Assume that the potential pair (q, r) belongs to the Schwartz class S(R). Then, we
have the following:

(a) The Jost solution ψ(ζ, x) to (1.6) is related to the Jost solution ψ̃(ζ, x) to (1.9) as[
ψ1(ζ, x)
ψ2(ζ, x)

]
= e−iµ/2

[
1 0
0 E(x)

] [
ψ̃1(ζ, x)

ψ̃2(ζ, x)

]
, (2.18)

where ψ̃1(ζ, x) and ψ̃2(ζ, x) denote the first and second components of the Jost solution

ψ̃(ζ, x) in a manner similar to the first equality of (2.1).

(b) The Jost solution ψ̄(ζ, x) to (1.6) is related to the Jost solution ˜̄ψ(ζ, x) to (1.9) as[
ψ̄1(ζ, x)
ψ̄2(ζ, x)

]
=

[
1 0
0 E(x)

][ ˜̄ψ1(ζ, x)
˜̄ψ2(ζ, x)

]
, (2.19)

where ˜̄ψ1(ζ, x) and ˜̄ψ2(ζ, x) denote the first and second components of the Jost solution
˜̄ψ(ζ, x) in a manner similar to the second equality of (2.1).

(c) The Jost solution ϕ(ζ, x) to (1.6) is related to the Jost solution ϕ̃(ζ, x) to (1.9) as[
ϕ1(ζ, x)
ϕ2(ζ, x)

]
=

[
1 0
0 E(x)

] [
ϕ̃1(ζ, x)

ϕ̃2(ζ, x)

]
, (2.20)

where ϕ̃1(ζ, x) and ϕ̃2(ζ, x) denote the first and second components of the Jost solution

ϕ̃(ζ, x) in a manner similar to the first equality of (2.2).

(d) The Jost solution ϕ̄(ζ, x) to (1.6) is related to the Jost solution ˜̄ϕ(ζ, x) to (1.9) as[
ϕ̄1(ζ, x)
ϕ̄2(ζ, x)

]
=

[
1 0
0 E(x)

][ ˜̄ϕ1(ζ, x)
˜̄ϕ2(ζ, x)

]
, (2.21)

where ˜̄ϕ1(ζ, x) and ˜̄ϕ2(ζ, x) denote the first and second components of the Jost solution
˜̄ϕ(ζ, x) in a manner similar to the second equality of (2.2).



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 7

Proof. We apply the relationship to (1.10) to the corresponding Jost solutions to (1.6) and (1.9),
respectively, and in each case we determine the specific value of the constant c appearing in (1.10)
for each pair of the Jost solutions. For example, to establish (2.18) we proceed as follows. We

write (1.10) by using the corresponding Jost solutions ψ(ζ, x) and ψ̃(ζ, x) there. This yields[
ψ1(ζ, x)
ψ2(ζ, x)

]
= c

[
1 0
0 E(x)

] [
ψ̃1(ζ, x)

ψ̃2(ζ, x)

]
. (2.22)

By letting x→ +∞ in (2.22), with the help of the asymptotics in (2.3) and (2.14) and the second
asymptotics in (1.12), we obtain c as e−iµ/2. The relationships presented in (b), (c), and (d) are
obtained in a similar manner with the help of the spacial asymptotics in (1.12), (2.4)–(2.6), and
(2.15)–(2.17). □

In the next theorem, we present the large ζ-asymptotics of the Jost solutions to (1.6). In the
theorem, those asymptotics are expressed in terms of λ, which is related to ζ as in (2.7).

Theorem 2.4. Assume that the potentials q and r in (1.6) belong to the Schwartz class S(R).
Let the parameter λ be related to the spectral parameter ζ as in (2.7). Then, for each fixed x ∈ R,
as λ → ∞ in C+ the Jost solutions ψ(ζ, x) and ϕ(ζ, x) to (1.6) appearing in (2.3) and (2.5),
respectively, satisfy

ψ1(ζ, x)

ζ
= eiλx

[q(x)
2iλ

+O
( 1

λ2

)]
, (2.23)

ψ2(ζ, x) = eiλx
[
1 +

q(x) r(x)

4λ
+

1

4λ

∫ ∞

x

dy q(y) r′(y) +O
( 1

λ2

)]
, (2.24)

ϕ1(ζ, x) = e−iλxE(x)
[
1 +

1

4λ

∫ x

−∞
dy q(y) r′(y) +O

( 1

λ2

)]
, (2.25)

ϕ2(ζ, x)

ζ
= e−iλxE(x)

[ i r(x)
2λ

+O
( 1

λ2

)]
, (2.26)

where we recall that E(x) and µ are the quantities in (1.11) and (1.13), respectively. Similarly,

for each fixed x ∈ R, as λ→ ∞ in C− the Jost solutions ψ̄(ζ, x) and ϕ̄(ζ, x) to (1.6) appearing in
(2.4) and (2.6), respectively, satisfy

ψ̄1(ζ, x) = e−iµ/2−iλxE(x)
[
1− 1

4λ

∫ ∞

x

dy q(y) r′(y) +O
( 1

λ2

)]
, (2.27)

ψ̄2(ζ, x)

ζ
= e−iµ/2−iλxE(x)

[ i r(x)
2λ

+O
( 1

λ2

)]
, (2.28)

ϕ̄1(ζ, x)

ζ
= eiλx

[q(x)
2iλ

+O
( 1

λ2

)]
, (2.29)

ϕ̄2(ζ, x) = eiλx
[
1 +

q(x) r(x)

4λ
− 1

4λ

∫ x

−∞
dy q(y) r′(y) +O

( 1

λ2

)]
. (2.30)

Proof. The large ζ-asymptotics of the Jost solutions to (1.9) are already known [12, 13] and they
are listed in Theorem 2.4 in [12]. We use those asymptotics on the right-hand sides of (2.18)–
(2.21). We also express q̃ and r̃ appearing in those asymptotics in terms of q and r with the help of
(2.8). Furthermore, we use (2.7) to relate the parameters λ and ζ to each other. We then obtain
the large ζ-asymptotics expressed in (2.23)–(2.30). □

Next, we introduce the scattering coefficients associated with the linear system (1.6) by using
the spacial asymptotics of the Jost solutions to (1.6) as[

ψ1(ζ, x)
ψ2(ζ, x)

]
=

 L(ζ)
Tl(ζ)

e−iζ2x
[
1 + o(1)

]
1

Tl(ζ)
eiζ

2x
[
1 + o(1)

]  , x→ −∞, (2.31)



8 M. UNLU EJDE-2025/97

[
ψ̄1(ζ, x)
ψ̄2(ζ, x)

]
=

 1
T̄l(ζ)

e−iζ2x
[
1 + o(1)

]
L̄(ζ)
T̄l(ζ)

eiζ
2x
[
1 + o(1)

]  , x→ −∞, (2.32)

[
ϕ1(ζ, x)
ϕ2(ζ, x)

]
=

 1
Tr(ζ)

e−iζ2x
[
1 + o(1)

]
R(ζ)
Tr(ζ)

eiζ
2x
[
1 + o(1)

]  , x→ +∞, (2.33)

[
ϕ̄1(ζ, x)
ϕ̄2(ζ, x)

]
=

 R̄(ζ)
T̄r(ζ)

e−iζ2x
[
1 + o(1)

]
1

T̄r(ζ)
eiζ

2x
[
1 + o(1)

]  , x→ +∞. (2.34)

We refer to Tl(ζ) and T̄l(ζ) as the left transmission coefficients, L(ζ) and L̄(ζ) as the left reflection
coefficients, Tr(ζ) and T̄r(ζ) as the right transmission coefficients, and R(ζ) and R̄(ζ) as the right
reflection coefficients.

Similarly, we introduce the scattering coefficients associated with the linear system (1.9) by
using the spacial asymptotics of the Jost solutions to (1.9) as[

ψ̃1(ζ, x)

ψ̃2(ζ, x)

]
=

 L̃(ζ)

T̃ (ζ)
e−iζ2x

[
1 + o(1)

]
1

T̃ (ζ)
eiζ

2x
[
1 + o(1)

]  , x→ −∞, (2.35)

[
˜̄ψ1(ζ, x)
˜̄ψ2(ζ, x)

]
=

 1
˜̄T (ζ)

e−iζ2x
[
1 + o(1)

]
˜̄L(ζ)
˜̄T (ζ)

eiζ
2x
[
1 + o(1)

]  , x→ −∞, (2.36)

[
ϕ̃1(ζ, x)

ϕ̃2(ζ, x)

]
=

 1
T̃ (ζ)

e−iζ2x
[
1 + o(1)

]
R̃(ζ)

T̃ (ζ)
eiζ

2x
[
1 + o(1)

]  , x→ +∞, (2.37)

[
˜̄ϕ1(ζ, x)
˜̄ϕ2(ζ, x)

]
=

 ˜̄R(ζ)
˜̄T (ζ)

e−iζ2x
[
1 + o(1)

]
1

˜̄T (ζ)
eiζ

2x
[
1 + o(1)

]  , x→ +∞. (2.38)

We refer to T̃ (ζ) and ˜̄T (ζ) as the transmission coefficients, L(ζ) and L̄(ζ) as the left reflection
coefficients, and R(ζ) and R̄(ζ) as the right reflection coefficients. We remark that we do not
need to make a distinction between the left and right transmission coefficients for (1.9). This is

due to the fact that the trace of the matrix X̃ appearing in (1.9) is zero. Consequently, the left
and right transmission coefficients for (1.9) coincide, and we use T (ζ) and T̄ (ζ) to denote those
common values, respectively. On the other hand, the trace of the matrix X appearing in (1.6)
is not zero. This results in the fact that the left and right transmission coefficients are unequal.
Hence, to describe the transmission coefficients for (1.6), we need to use the four quantities Tl(ζ),
Tr(ζ), T̄l(ζ), and T̄r(ζ).

In the next theorem, we show the connection among the scattering coefficients for (1.6) and the
scattering coefficients for (1.9).

Theorem 2.5. Let the potential pair (q, r) in (1.6) and the potential pair (q̃, r̃) in (1.9) are related
to each other as in (2.8), where µ is the quantity appearing in (1.13). Assume that the potential
pair (q, r) belongs to the Schwartz class S(R). Then, we have the following:

(a) The eight scattering coefficients Tl(ζ), Tr(ζ), T̄l(ζ), T̄r(ζ), R(ζ), L(ζ), R̄(ζ), L̄(ζ) for (1.6)

are related to the six scattering coefficients T̃ (ζ), ˜̄T (ζ), R̃(ζ), L̃(ζ), ˜̄R(ζ), ˜̄L(ζ) for (1.9) as

Tl(ζ) = eiµ/2 T̃ (ζ), T̄l(ζ) =
˜̄T (ζ), (2.39)

Tr(ζ) = T̃ (ζ), T̄r(ζ) = e−iµ/2 ˜̄T (ζ), (2.40)

R(ζ) = eiµ/2 R̃(ζ), R̄(ζ) = e−iµ/2 ˜̄R(ζ), (2.41)

L(ζ) = L̃(ζ), L̄(ζ) = ˜̄L(ζ). (2.42)



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 9

(b) The transmission coefficients Tl(ζ) and Tr(ζ) are even in ζ, and hence they are functions of
λ. As functions of λ, the quantities Tl(ζ) and Tr(ζ) are meromorphic in λ ∈ C+ and continuous in

λ ∈ C+ except at the poles causing the meromorphic property in C+. Similarly, the transmission
coefficients T̄l(ζ) and T̄r(ζ) are even in ζ, and hence they are functions of λ. As functions of λ,

the quantities T̄l(ζ) and T̄r(ζ) are meromorphic in λ ∈ C− and continuous in λ ∈ C− except at
the poles causing the meromorphic property in C−. The four quantities R(ζ)/ζ, R̄(ζ)/ζ, L(ζ)/ζ,
L̄(ζ)/ζ are even in ζ, and hence they are all functions of λ. As functions of λ, those four quantities
are continuous in λ ∈ R.

Proof. For the proof of (a), we use the relationships (2.18)–(2.21) connecting to Jost solutions to
(1.6) and the Jost solutions to (1.9). With the help of the asymptotics in (1.12) and (2.31)–(2.38),
by comparing the leading terms in the spacial asymptotics of (2.18)–(2.21), we obtain (2.39)–
(2.42). Hence, the proof of (a) is complete. By using the relevant properties of the scattering
coefficients T (ζ), T̄ (ζ), L(ζ), L̄(ζ), R(ζ), and R̄(ζ) given in Theorem 2.2 of [12], we establish the
aforementioned properties of the scattering coefficients for (1.6). □

In the next theorem, we present the small ζ-asymptotics of the scattering coefficients for (1.6).

Theorem 2.6. Assume that the potentials q and r in (1.6) belong to the Schwartz class S(R) in
x ∈ R. Let the parameter λ be related to the spectral parameter ζ as in (2.7). Then, the small
ζ-asymptotics of the scattering coefficients Tl(ζ), Tr(ζ), T̄l(ζ), T̄r(ζ), R(ζ), R̄(ζ), L(ζ), and L̄(ζ)
appearing in (2.31)–(2.34) are expressed in λ as

Tl(ζ) = eiµ/2
[
1 +O(λ)

]
, λ→ 0 in C+, (2.43)

Tr(ζ) = 1 +O(λ), λ→ 0 in C+, (2.44)

T̄l(ζ) = 1 +O(λ), λ→ 0 in C+, (2.45)

T̄r(ζ) = e−iµ/2
[
1 +O(λ)

]
, λ→ 0 in C+, (2.46)

R(ζ)

ζ
= eiµ/2

[ 1

E(x)

∫ ∞

−∞
dy r(y) +O(λ)

]
, λ→ 0 in R, (2.47)

R̄(ζ)

ζ
= e−iµ/2

[
E(x)

∫ ∞

−∞
dy q(y) +O(λ)

]
, λ→ 0 in R, (2.48)

L(ζ)

ζ
= −

[
E(x)

∫ ∞

−∞
dy q(y) +O(λ)

]
, λ→ 0 in R, (2.49)

L̄(ζ)

ζ
= −

[ 1

E(x)

∫ ∞

−∞
dy r(y) +O(λ)

]
, λ→ 0 in R. (2.50)

Proof. To obtain (2.43)–(2.50), we use the connections among the scattering coefficients for (1.6)
and for (1.9) given in (2.39)–(2.42). We utilize the help of (2.8) and the known small ζ asymptotics
of the scattering coefficients for (1.9), where those asymptotics are listed in Theorem 2.5 (d) of
[12]. We then obtain (2.43)–(2.50). □

In the next theorem, we present the large ζ-asymptotics of the scattering coefficients for (1.6).
In the theorem, those asymptotics are expressed in terms of λ, which is related to ζ as in (2.7).

Theorem 2.7. Assume that the potentials q and r in (1.6) belong to the Schwartz class S(R) in
x ∈ R. Let the parameter λ be related to the spectral parameter ζ as in (2.7). Then, the large
ζ-asymptotics of the scattering coefficients Tl(ζ), Tr(ζ), T̄l(ζ), T̄r(ζ), R(ζ), R̄(ζ), L(ζ), and L̄(ζ)
appearing in (2.31)–(2.34) are expressed in λ as

Tl(ζ) = 1 +O
( 1
λ

)
, λ→ ∞ in C+, (2.51)

T̄l(ζ) = eiµ/2
[
1 +O

( 1
λ

)]
, λ→ ∞ in C+, (2.52)

Tr(ζ) = e−iµ/2
[
1 +O

( 1
λ

)]
, λ→ ∞ in C+, (2.53)



10 M. UNLU EJDE-2025/97

T̄r(ζ) = 1 +O
( 1

λ

)
, λ→ ∞ in C+, (2.54)

R(ζ) = O
( 1

ζ3

)
, R̄(ζ) = O

( 1

ζ3

)
, λ→ ±∞, (2.55)

L(ζ) = O
( 1

ζ3

)
, L̄(ζ) = O

( 1

ζ3

)
, λ→ ±∞. (2.56)

Proof. The large ζ-asymptotics of the scattering coefficients for (1.9) are known and they are listed
in (2.46)–(2.51) of [12]. Using those asymptotics in (2.39)–(2.42), we obtain (2.51)–(2.56). □

In the next theorem we show that the left scattering coefficients Tl(ζ), T̄l(ζ), L(ζ), L̄(ζ) can be
expressed in terms of the right scattering coefficients Tr(ζ), T̄r(ζ), R(ζ), R̄(ζ). Hence, in solving the
inverse scattering problem for (1.6), instead of using all the scattering coefficients, it is sufficient to
use as input a scattering data set containing the right scattering coefficients but not left scattering
coefficients.

Theorem 2.8. Assume that the potentials q and r in (1.6) belong to the Schwartz class S(R) in
x ∈ R. The left scattering coefficients Tl(ζ), T̄l(ζ), L(ζ), L̄(ζ) for (1.6) can be expressed in terms
of the right scattering coefficients Tr(ζ), T̄r(ζ), R(ζ), R̄(ζ) for (1.6) as

Tl(ζ) = eiµ/2Tr(ζ), T̄l(ζ) = eiµ/2T̄r(ζ), (2.57)

L(ζ) = −eiµ/2R̄(ζ)Tr(ζ)
T̄r(ζ)

, L̄(ζ) = −R(ζ) T̄r(ζ)
Tr(ζ)

, (2.58)

where µ is the complex constant defined in (1.13).

Proof. We obtain the first equality in (2.57) directly from the first equalities of (2.39) and (2.40).
Similarly, the second equality in (2.57) is obtained by using the second equalities in (2.39) and
(2.40). The analogs of the first and the second equalities in (2.58) for the linear system (1.9) are
known from [12, (2.45)] and we have

L̃(ζ) = −eiµ/2 ˜̄R(ζ) T̃r(ζ)
˜̄Tr(ζ)

, ˜̄L(ζ) = −R̃(ζ)
˜̄Tr(ζ)

T̃r(ζ)
. (2.59)

By using the relationships between the reflection coefficients for (1.6) and the reflection coefficients
for (1.9) in (2.59), namely, using (2.59) in (2.42), we obtain the first and the second equalities in
(2.58), respectively. □

3. Bound states

We recall that the potential pair (q, r) appearing in (1.6) is assumed to belong to the Schwartz
class S(R) in x ∈ R. The bound states for (1.6) correspond to square integrable column-vector
solutions to (1.6). When the spectral parameter ζ is real, by using any two linearly independent
column-vector solutions to (1.6), it is impossible to form a square integrable solution to (1.6).
Hence, there are no bound states for (1.6) when ζ ∈ R. A bound state for (1.6) can only occur at
a nonreal complex value of ζ.

As indicated in Theorem 2.2, the potential pair (q̃, r̃) in (1.9) belongs to the Schwartz class
S(R) when the potential pair (q, r) in (1.6) belongs S(R). The bound states for (1.6) are closely
related to the meromorphic properties of the transmission coefficients in the complex ζ-plane.
From (2.40) and (2.41) we know that the transmission coefficients for (1.6) and the transmission
coefficients for (1.9) have similar meromorphic properties in the complex ζ-plane. Thus, by using
the facts outlined in Section 3 of [12] for the bound states for (1.9), we obtain the facts related to
the bound states for (1.6). We refer the reader to Section 3 of [12] for the details about the bound
states for (1.9), and in the following we provide a summary related to the bound states for (1.6)
when the potential pair (q, r) belongs to the Schwartz class.

(a) As already indicated, the bound states for (1.6) cannot occur when ζ ∈ R. A bound state
can only occur at a nonreal complex ζ-value at which the transmission coefficient Tr(ζ) has a pole
in the first or third quadrant in the complex ζ-plane or the transmission coefficient T̄r(ζ) has a pole



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 11

in the second or fourth quadrant. As a consequence of (2.57), the poles and their multiplicities
for Tl(ζ) and Tr(ζ) coincide and the poles and their multiplicities for T̄l(ζ) and T̄r(ζ) coincide.
From Theorem 2.5(b) we know that the transmission coefficients Tr(ζ) and T̄r(ζ) are even in ζ,
and hence the ζ-values corresponding to the bound states for (1.6) are located symmetrically with
respect to the origin of the complex ζ-plane. Thus, it is convenient to describe the bound-state
poles of Tr(ζ) and T̄r(ζ) in terms of the parameter λ related to ζ as in (2.7).

(b) The number of poles of Tr(ζ) in the upper-half complex λ-plane is finite, and we use λj
for 1 ≤ j ≤ N to denote those distinct poles of Tr(ζ). Similarly, the number of poles of T̄r(ζ) in
the lower-half complex λ-plane is finite, and we use λ̄j for 1 ≤ j ≤ N̄ to denote those distinct λ̄j
values. It is possible that Tr(ζ) has no poles in the upper-half complex λ-plane, in which case we
have N = 0. Similarly, it is possible that T̄r(ζ) has no poles in the lower-half complex λ-plane, in
which case we have N̄ = 0. The multiplicity of the pole of Tr(ζ) at λ = λj is finite, and we use
mj to denote that multiplicity. Similarly, the multiplicity of the pole of T̄r(ζ) at λ = λ̄j is finite,
and we use m̄j to denote that multiplicity.

(c) The bound-state information for (1.6) contains the sets {λj ,mj}Nj=1 and
{
λ̄j , m̄j

}N̄

j=1
. For

each bound state and multiplicity, we specify a bound-state normalization constant. To denote
the bound-state normalization constants, we use the double-indexed quantities cjk for 1 ≤ j ≤ N
and 0 ≤ k ≤ mj − 1 and the double-indexed quantities c̄jk for 1 ≤ j ≤ N̄ and 0 ≤ k ≤ m̄j − 1.
The construction of the complex-valued normalization constants cjk and c̄jk for (1.6) is similar to
the construction given in [11, 12] of the bound-state normalization constants for (1.9). Thus, the
bound-state information for (1.6) consists of the two sets given by{

λj ,mj , {cjk}
mj−1
k=0

}N

j=1
,

{
λ̄j , m̄j , {c̄jk}

m̄j−1
k=0

}N̄

j=1
. (3.1)

We refer the reader to Examples 6.1 and 6.2 in [12], where it is illustrated how the bound-state
normalization constants for (1.9) constructed by using the transmission coefficients and the bound-
state dependency constants for (1.9).

(d) The bound-state information presented in (3.1) can be organized by using a pair of matrix
triplets. We use the matrix triplet (A,B,C) and the matrix triplet (Ā, B̄, C̄) to represent the
information contained in the first and the second sets, respectively, appearing in (3.1). The
specification of the bound-state information in the form of a pair of matrix triplets is especially
convenient in the solution of the inverse scattering problem for (1.6) with the help of the solution
to a Marchenko system of integral equations. The advantage of using matrix triplets to represent
the bound-state information is that it allows us to deal with any number of bound states with any
multiplicities as if we deal only with a single bound state having the multiplicity equal to 1.

The translation of the bound-state information from (3.1) to the matrix triplets (A,B,C) and
(Ā, B̄, C̄) is as follows. For each bound state at λ = λj for 1 ≤ j ≤ N with the multiplicity mj ,
we form the matrix subtriplet (Aj , Bj , Cj) by letting

Aj :=



λj 1 0 · · · 0 0
0 λj 1 · · · 0 0
0 0 λj · · · 0 0
...

...
...

. . .
...

...
0 0 0 · · · λj 1
0 0 0 . . . 0 λj


, (3.2)

Bj :=


0
...
0
1

 , Cj :=
[
cj(mj−1) cj(mj−2) · · · cj1 cj0

]
. (3.3)

Here, Aj is themj×mj square matrix in the Jordan canonical form with λj in the diagonal entries,
Bj is the column vector with mj components that are all zero with the exception of the last entry
being 1, and Cj is the row vector with mj components containing the bound-state normalization



12 M. UNLU EJDE-2025/97

constants cjk for 0 ≤ k ≤ mj − 1 in the order indicated in (3.3). In a similar manner, for each
bound state at λ = λ̄j for 1 ≤ j ≤ N̄ , we form the matrix subtriplet (Āj , B̄j , C̄j) as

Āj :=



λ̄j 1 0 · · · 0 0
0 λ̄j 1 · · · 0 0
0 0 λ̄j · · · 0 0
...

...
...

. . .
...

...
0 0 0 · · · λ̄j 1
0 0 0 . . . 0 λ̄j


, (3.4)

B̄j :=


0
...
0
1

 , C̄j :=
[
c̄j(m̄j−1) c̄j(m̄j−2) · · · c̄j1 c̄j0

]
. (3.5)

We note that Āj is the m̄j×m̄j square matrix in the Jordan canonical form with λ̄j in the diagonal
entries, B̄j is the column vector with the first m̄j − 1 entries being zero and with the last entry
being 1, and C̄j is the row vector with m̄j components containing the bound-state normalization
constants c̄jk for 0 ≤ k ≤ m̄j − 1 in the order indicated in (3.5).

By using the subtriplets (Aj , Bj , Cj) and (Āj , B̄j , C̄j) given in (3.2)–(3.5), we transform the
bound-state information from (3.1) to the matrix triplets (A,B,C) and (Ā, B̄, C̄) by letting

A :=


A1 0 · · · 0 0
0 A2 · · · 0 0
...

...
. . .

...
...

0 0 · · · AN−1 0
0 0 · · · 0 AN

 , Ā :=


Ā1 0 · · · 0 0
0 Ā2 · · · 0 0
...

...
. . .

...
...

0 0 · · · ĀN̄−1 0
0 0 · · · 0 ĀN̄

 , (3.6)

B :=


B1

B2

...
BN

 , B̄ :=


B̄1

B̄2

...
B̄N̄

 , (3.7)

C :=
[
C1 C2 · · · CN

]
, C̄ :=

[
C̄1 C̄2 · · · C̄N̄

]
. (3.8)

We remark that the matrices A, B, C, Ā, B̄, C̄ each are block matrices, and the zeros in (3.6)
denote the zero matrices of appropriate matrix sizes. The matrix size of A is N × N and the
matrix size of Ā is N̄ × N̄ , where we have defined

N :=

N∑
j=1

mj , N̄ :=

N̄∑
j=1

m̄j . (3.9)

The matrices B and B̄ are column vectors with N and N̄ components, respectively. Similarly, the
matrices C and C̄ are row vectors with N and N̄ components, respectively.

4. The Marchenko method

In this section we present the Marchenko method [5, 26] for (1.6) by deriving the corresponding
Marchenko system of linear integral equations. In the Marchenko method, the potentials q and
r are recovered from the input scattering data set consisting of the scattering coefficients and
the bound-state information. The input is used to construct the kernel in the Marchenko system
of linear integral equations as well as the nonhomogeneous term in the Marchenko system. The
potentials and all other relevant quantities associated with (1.6) are then recovered from the
solution to the Marchenko system.

In the next theorem, we present the derivation of the Marchenko system of integral equations
for (1.6) in the absence of bound states.



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 13

Theorem 4.1. Assume that the potentials q and r in (1.6) belong to the Schwartz class S(R) in
x ∈ R. In the absence of bound states, the Marchenko system of linear integral equations for (1.6)
is [

0 0
0 0

]
=

[
K̄1(x, y) K1(x, y)
K̄2(x, y) K2(x, y)

]
+

[
0 ˆ̄R(x+ y)

R̂(x+ y) 0

]

+

∫ ∞

x

dz

[
−iK1(x, z)R̂

′(z + y) K̄1(x, z)
ˆ̄R(z + y)

K2(x, z)R̂(z + y) iK̄2(x, z)
ˆ̄R′(z + y)

]
, x < y,

(4.1)

where R̂(y) and ˆ̄R(y) are related to the reflection coefficients R(ζ) and R̄(ζ) for (1.6) via the
Fourier transforms given by

R̂(y) :=
1

2π

∫ ∞

−∞
dλ
R(ζ)

ζ
eiλy, ˆ̄R(y) :=

1

2π

∫ ∞

−∞
dλ
R̄(ζ)

ζ
e−iλy, (4.2)

with R̂′(y) and ˆ̄R′(y) denoting the derivatives of R̂(y) and ˆ̄R(y), respectively, and λ being the
parameter related to ζ as in (2.7). The quantities K1(x, y), K2(x, y), K̄1(x, y), and K̄2(x, y) are
related to the components of the Jost solutions ψ(ζ, x) and ψ̄(ζ, x) appearing in (2.1) as

K1(x, y) :=
1

2π

∫ ∞

−∞
dλ

[eiµ/2 ψ1(ζ, x)

ζ E(x)

]
e−iλy, (4.3)

K2(x, y) :=
1

2π

∫ ∞

−∞
dλ

[
ψ2(ζ, x)− eiλx

]
e−iλy, (4.4)

K̄1(x, y) :=
1

2π

∫ ∞

−∞
dλ

[ eiµ/2
E(x)

ψ̄1(ζ, x)− e−iλx
]
eiλy, (4.5)

K̄2(x, y) :=
1

2π

∫ ∞

−∞
dλ

[ ψ̄2(ζ, x)

ζ

]
eiλy, (4.6)

with E(x) and µ being the quantities defined in (1.11) and (1.13), respectively.

Proof. For notational simplicity in the derivation of (4.1), we suppress the arguments and write
ψ for ψ(ζ, x), ψ̄ for ψ̄(ζ, x), ϕ for ϕ(ζ, x), ϕ̄ for ϕ̄(ζ, x), Tr for Tr(ζ), T̄r for T̄r(ζ), R for R(ζ), R̄
for R̄(ζ), and E for E(x). From the asymptotics in (2.3) and (2.4) we see that the column-vector
Jost solutions ψ and ψ̄ to (1.6) are linearly independent, and hence those four column-vector
solutions form a fundamental set of column-vector solutions to (1.6). Thus, each of the other two
column-vector Jost solutions ϕ and ϕ̄ can be expressed as linear combinations of ψ and ψ̄. With
the help of (2.3), (2.4), (2.33), and (2.34), for ζ ∈ R we obtain

ϕ =
1

Tr
ψ̄ +

R

Tr
ψ,

ϕ̄ =
R̄

T̄r
ψ̄ +

1

T̄r
ψ,

or equivalently
Tr ϕ = ψ̄ +Rψ,

T̄rϕ̄ = R̄ψ̄ + ψ,
(4.7)

which forms a basis for our Riemann–Hilbert problem. The solution to the Riemann–Hilbert
problem consists of the construction of the Jost solutions from the knowledge of the scattering
coefficients Tr, T̄r, R, R̄ appearing as coefficients in (4.7). We derive our Marchenko system of
integral equations starting from (4.7) by proceeding as follows. We first combine the two column-
vector equations in (4.7) and obtain the 2× 2 matrix-valued system given by[

Tr ϕ T̄rϕ̄
]
=

[
ψ̄ ψ

]
+
[
Rψ R̄ψ̄

]
. (4.8)

Using (2.1) and (2.2), we write (4.8) as[
Tr ϕ1 T̄rϕ̄1
Tr ϕ2 T̄rϕ̄2

]
=

[
ψ̄1 ψ1

ψ̄2 ψ2

]
+

[
Rψ1 R̄ψ̄1

Rψ2 R̄ψ̄2

]
. (4.9)



14 M. UNLU EJDE-2025/97

Premultiplying both sides of (4.9) by the 2× 2 matrix

[
eiµ/2E−1 0

0 1

]
, we obtain[

eiµ/2E−1Tr ϕ1 eiµ/2E−1T̄rϕ̄1
Tr ϕ2 T̄rϕ̄2

]
=

[
eiµ/2E−1ψ̄1 eiµ/2E−1ψ1

ψ̄2 ψ2

]
+

[
eiµ/2E−1Rψ1 eiµ/2E−1R̄ψ̄1

Rψ2 R̄ψ̄2

]
.

(4.10)

Subtracting the 2 × 2 matrix

[
e−iλx 0
0 eiλx

]
from each side of (4.10) and then dividing the off-

diagonal entries by ζ in the resulting matrix equality, we obtain[
eiµ/2E−1Trϕ1 − e−iλx eiµ/2

ζ E−1T̄rϕ̄1
1
ζTrϕ2 T̄rϕ̄2 − eiλx

]
=

[
eiµ/2E−1ψ̄1 − e−iλx eiµ/2

ζ E−1ψ1
1
ζ ψ̄2 ψ2 − eiλx

]

+

[
eiµ/2E−1Rψ1

eiµ/2

ζ E−1R̄ψ̄1
1
ζRψ2 R̄ψ̄2

]
.

(4.11)

Next, we take the Fourier transform of (4.11) with
∫∞
−∞ dλeiλy/2π in the first columns and with∫∞

−∞ dλe−iλy/2π in the second columns. This yields the 2× 2 matrix-valued equation

LHS = K(x, y) + RHS, (4.12)

where we have defined

K(x, y) :=

[
K̄1(x, y) K1(x, y)
K̄2(x, y) K2(x, y)

]
, (4.13)

with the entries K1(x, y), K2(x, y), K̄1(x, y), and K̄2(x, y) are as in (4.3)–(4.6), respectively. In
(4.12) we have

LHS :=

[
LHS11 LHS12
LHS21 LHS22

]
, (4.14)

RHS :=

[
RHS11 RHS12
RHS21 RHS22

]
, (4.15)

with the matrix entries in (4.14) and (4.15) defined as

LHS11 :=

∫ ∞

−∞

dλ

2π

[ eiµ/2
E(x)

Tr(ζ)ϕ1(ζ, x)− e−iλx
]
eiλy, (4.16)

LHS12 :=

∫ ∞

−∞

dλ

2π

[ eiµ/2
ζE(x)

T̄r(ζ)ϕ̄1(ζ, x)
]
e−iλy, (4.17)

LHS21 :=

∫ ∞

−∞

dλ

2π

[1
ζ
Tr(ζ)ϕ2(ζ, x)

]
eiλy, (4.18)

LHS22 :=

∫ ∞

−∞

dλ

2π

[
T̄r(ζ)ϕ̄2(ζ, x)− eiλx

]
e−iλy, (4.19)

RHS11 :=

∫ ∞

−∞

dλ

2π

[ eiµ/2
E(x)

R(ζ)ψ1(ζ, x)
]
eiλy, (4.20)

RHS12 :=

∫ ∞

−∞

dλ

2π

[ eiµ/2
ζE(x)

R̄(ζ)ψ̄1(ζ, x)
]
e−iλy, (4.21)

RHS21 :=

∫ ∞

−∞

dλ

2π

[1
ζ
R(ζ)ψ2(ζ, x)

]
eiλy, (4.22)

RHS22 :=

∫ ∞

−∞

dλ

2π

[
R̄(ζ)ψ̄2(ζ, x)

]
e−iλy. (4.23)

In the absence of bound states, when x < y the integrands in (4.3) and (4.4) are analytic in

λ ∈ C+, are continuous in λ ∈ C+, and behave as eiλ(y−x)O(1/λ) as λ → ∞ in C+. Similarly, in
the absence of bound states, when x < y the integrands in (4.5) and (4.6) are analytic in λ ∈ C−,



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 15

are continuous in λ ∈ C−, and behave as e−iλ(y−x)O(1/λ) as λ→ ∞ in C−. Thus, from Jordan’s
lemma it follows that the matrix K(x, y) defined in (4.11) is equal to zero when x > y. On the
other hand, in the absence of bound states, when x < y with the help of Theorems 2.1, 2.4, and
2.6 we observe that the integrands in (4.16) and (4.18) are analytic in λ ∈ C+, are continuous

in λ ∈ C+, and behave uniformly as O(1/λ) as λ → ∞ in C+. Similarly, when x < y, with
the help of Theorems 2.1, 2.4, and 2.6, we observe that the integrands in (4.17) and (4.19) are

analytic in λ ∈ C−, continuous in λ ∈ C−, and behave as O(1/λ) as λ → ∞ in C−. Hence,
when x < y, using Jordan’s lemma and the residue theorem, we conclude that the matrix LHS
defined in (4.14) is zero. In fact, from the continuity of the Jost solutions stated in Theorem 2.1,
the continuity and asymptotic properties of the scattering coefficients stated in Theorem 2.7, the
large ζ-asymptotics of the Jost solutions stated in Theorems 2.4, we observe that each integrand
in (4.3)–(4.6) and (4.16)–(4.23) is continuous in λ ∈ R and behaves as O(1/λ) as λ→ ±∞. Thus,
the L2-Fourier transforms in (4.3)–(4.6) and (4.16)–(4.23) are all well defined. Using the inverse
Fourier transform, from (4.3)–(4.6) we obtain

eiµ/2

ζE(x)
ψ1(ζ, x) =

∫ ∞

x

dyK1(x, y)e
iλy, (4.24)

ψ2(ζ, x) = eiλx +

∫ ∞

x

dyK2(x, y)e
iλy, (4.25)

eiµ/2

E(x)
ψ̄1(ζ, x) = e−iλx +

∫ ∞

x

dy K̄1(x, y)e
−iλy, (4.26)

1

ζ
ψ̄2(ζ, x) =

∫ ∞

x

dy K̄2(x, y)e
−iλy. (4.27)

Form (4.2), by using the inverse Fourier transform we have

R(ζ)

ζ
=

∫ ∞

−∞
ds R̂(s)e−iλs,

R̄(ζ)

ζ
=

∫ ∞

−∞
ds ˆ̄R(s)eiλs. (4.28)

Taking the y-derivatives, from (4.2) we obtain

R̂′(y) =
i

2π

∫ ∞

−∞
dλ

R(ζ)

ζ
λeiλy, ˆ̄R′(y) = − i

2π

∫ ∞

−∞
dλ

R̄(ζ)

ζ
λe−iλy. (4.29)

Using the inverse Fourier transform, from (4.29) we obtain

R(ζ)

ζ
λ = −i

∫ ∞

−∞
ds R̂′(s)e−iλs,

R̄(ζ)

ζ
λ = i

∫ ∞

−∞
ds ˆ̄R′(s)eiλs. (4.30)

Next, we take the Fourier transform of each component of the matrix RHS appearing in (4.15).
For this, we first write (4.20) in an equivalent form, i.e.

RHS11 =

∫ ∞

−∞

dλ

2π
eiλy

( eiµ/2

ζE(x)
ψ1(ζ, x)

)(R(ζ)
ζ

λ
)
. (4.31)

Then, using (4.24) and the first equality of (4.30) on the right-hand side of (4.31), we obtain

RHS11 = −i
∫ ∞

x

dz K1(x, z)R̂
′(z + y), (4.32)

where we have used the fact that K1(x, z) vanishes when x > z. Similarly, we write (4.23) in an
equivalent form as

RHS22 =

∫ ∞

−∞

dλ

2π
e−iλy

(
e−iµ/2E(x)

ψ̄2(ζ, x)

ζ

)( R̄(ζ)
ζ

λ
)
. (4.33)

Then, using (4.27) and the second equality of (4.30) on the right-hand side of (4.33), we obtain

RHS22 = i

∫ ∞

x

dzK̄2(x, z)
ˆ̄R′(z + y), (4.34)



16 M. UNLU EJDE-2025/97

where we have used the fact that K̄2(x, z) vanishes when x > z. Proceeding in a similar manner,
by using (4.25), (4.26), and (4.28), we write (4.21) and (4.22), respectively, as

RHS12 = ˆ̄R(x+ y) +

∫ ∞

x

dz K̄1(x, z)
ˆ̄R(z + y), (4.35)

RHS21 = R̂(x+ y) +

∫ ∞

x

dz K2(x, z)R̂(z + y). (4.36)

Hence, using (4.32), (4.34), (4.35), and (4.36) in (4.12), we observe that RHS is equal to the sum
of the second and third terms on the right-hand side of (4.1). Thus, the proof is complete. □

When (1.6) has bound states, the only modification needed in the proof of Theorem 4.1 is that
the quantity LHS appearing in (4.12) is no longer equal to the zero matrix because of that we
must take into account the bound-state poles of the transmission coefficients Tr(ζ) and T̄r(ζ) in
evaluating the integrals in (4.16)–(4.19). Those integrals, after using the poles of Tr(ζ) and T̄r(ζ)
and the bound-state dependency constants for (1.6), can be explicitly evaluated and the results can
be expressed in terms of the matrix triplets (A,B,C) and (Ā, B̄, C̄) appearing in (3.6)–(3.8). This
yields the Marchenko system of integral equations presented in the next theorem in the presence
of bound states for (1.6). The proof of the theorem is analogous to the proof of [13, Theorem 4.2]
involving the derivation of the Marchenko system for (1.9) in the presence of bound states. Hence,
we omit the proof. In order to present the Marchenko system of integral equations for (1.6) in the
presence of bound states, we introduce the 2× 2 matrix-valued quantities Ω(y) and Ω̄(y) as

Ω(y) := R̂(y) + CeiAyB, Ω̄(y) := ˆ̄R(y) + C̄e−iĀyB̄, (4.37)

where we use eiAy and e−iĀy to denote the corresponding matrix exponentials. By taking the
y-derivative of the two matrix equalities in (4.37), we obtain

Ω′(y) = R̂′(y) + iCAeiAyB, Ω̄′(y) = ˆ̄R′(y)− iC̄Āe−iĀyB̄. (4.38)

Theorem 4.2. Assume that the potentials q and r in (1.6) belong to the Schwartz class S(R)
in x ∈ R. Let (A,B,C) and (Ā, B̄, C̄) be the pair of matrix triplets representing the bound-state
information for (1.6). In the presence of bound states, the Marchenko system for (1.6) is given by[

0 0
0 0

]
=

[
K̄1(x, y) K1(x, y)
K̄2(x, y) K2(x, y)

]
+

[
0 Ω̄(x+ y)

Ω(x+ y) 0

]
+

∫ ∞

x

dz

[
−iK1(x, z) Ω

′(z + y) K̄1(x, z)Ω̄(z + y)
K2(x, z) Ω(z + y) iK̄2(x, z)Ω̄

′(z + y)

]
, x < y,

(4.39)

where Ω(y), Ω̄(y), Ω′(y), Ω̄′(y) are the quantities appearing in (4.37) and (4.38), and K1(x, y),
K2(x, y), K̄1(x, y), K̄2(x, y) are the quantities appearing in (4.3)–(4.6), respectively.

Using the four equalities arising from the four entries in (4.39), we write the Marchenko system
as a coupled system of four integral equations holding for x < y as

K̄1(x, y)− i

∫ ∞

x

dz K1(x, z) Ω
′(z + y) = 0,

K1(x, y) + Ω̄(x+ y) +

∫ ∞

x

dz K̄1(x, z)Ω̄(z + y) = 0,

K̄2(x, y) + Ω(x+ y) +

∫ ∞

x

dz K2(x, z) Ω(z + y) = 0,

K2(x, y) + i

∫ ∞

x

dz K̄2(x, z)Ω̄
′(z + y) = 0.

(4.40)



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 17

We can uncouple the Marchenko system (4.40) by using the first line in the second equality and
by using the fourth line in the third equality. We obtain

K1(x, y) + Ω̄(x+ y) + i

∫ ∞

x

dz K1(x, z)

∫ ∞

x

dsΩ′(z + s)Ω̄(s+ y) = 0,

K̄2(x, y) + Ω(x+ y)− i

∫ ∞

x

dzK̄2(x, z)

∫ ∞

x

ds Ω̄′(z + s) Ω(s+ y) = 0,

(4.41)

and

K̄1(x, y) = i

∫ ∞

x

dz K1(x, z) Ω
′(z + y),

K2(x, y) = −i
∫ ∞

x

dz K̄2(x, z)Ω̄
′(z + y),

(4.42)

where it is understood that we first solve the two uncoupled integral equations given in (4.41) and
obtain K1(x, y) and K̄2(x, y) and then use those values in the integrands in (4.42) and recover
K̄1(x, y) and K2(x, y).

In the next theorem, we relate the quantities K1(x, x), K̄1(x, x), K2(x, x), K̄2(x, x) obtained
from the solutionK(x, y) to the Marchenko system (4.39) to some quantities related to the potential
pair (q, r) in (1.6).

Theorem 4.3. Suppose that the potentials q and r appearing in (1.6) belong to the Schwartz class
S(R) in x ∈ R. Let K(x, y) be the solution to the Marchenko system (4.39), with the components
K1(x, y), K2(x, y), K̄1(x, y), K̄2(x, y) as in (4.13). In the limit y → x+ we have

K1(x, x) = −e
iµ/2

2

q(x)

E(x)
, (4.43)

K2(x, x) = − iq(x) r(x)
4

− i

4

∫ ∞

x

dy q(y) r′(y), (4.44)

K̄1(x, x) =
1

2

∫ ∞

x

dy q(y) r′(y), (4.45)

K̄2(x, x) = −e
−iµ/2

2
r(x)E(x), (4.46)

where E(x) and µ are the quantities defined in (1.11) and (1.13), respectively, and for notational
simplicity we use K1(x, x), K2(x, x), K̄1(x, x), K̄2(x, x) for K1(x, x

+), K2(x, x
+), K̄1(x, x

+),
K̄2(x, x

+), respectively.

Proof. We rewrite (4.24) as

eiµ/2

ζE(x)
ψ1(ζ, x) =

∫ ∞

x

dy
[
K1(x, y)

d

dy

eiλy

iλ

]
, (4.47)

where we recall that the parameter λ is related to ζ as in (2.7). Using integration by parts in
(4.47), we obtain

eiµ/2 ψ1(ζ, x)

ζ E(x)
= −K1(x, x)

eiλx

iλ
−
∫ ∞

x

dy
eiλy

iλ

∂ K1(x, y)

∂y
, (4.48)

where we have used K1(x,+∞) = 0. Letting λ→ ∞ in C+, from (4.48) we obtain

eiµ/2ψ1(ζ, x)

ζ E(x)
= −K1(x, x)

eiλx

iλ
+O

( 1

λ2

)
. (4.49)

Using the large ζ-asymptotics of ψ1(ζ, x) given in (2.23) on the left-hand side of (4.49), we have

eiµ/2eiλx

E(x)

[q(x)
2iλ

+O
( 1

λ2

)]
= −K1(x, x)e

iλx

iλ
+O

( 1

λ2

)
, λ→ ∞ in C+. (4.50)

A comparison of the O(1/λ)-terms on both sides of (4.50) yields (4.43). The equalities (4.44)–
(4.46) are obtained in a similar manner. □



18 M. UNLU EJDE-2025/97

In the next theorem, we show how to recover the quantity E(x), the potential pair (q, r), and
the Jost solutions ψ(ζ, x) and ψ̄(ζ, x) to (1.6) from the solution K(x, y) to the Marchenko system
(4.39).

Theorem 4.4. Assume that the potentials q and r in (1.6) belong to the Schwartz class S(R) in
x ∈ R. The quantity E(x) in (1.11), the constant µ in (1.13), the potential pair (q, r), and the
Jost solutions ψ(ζ, x) and ψ̄(ζ, x) to (1.6) are recovered from the solution K(x, y) to the Marchenko
system (4.39) as follows:

(a) We have

E(x) = exp
(
2i

∫ x

−∞
dz P (z)

)
, µ = 4

∫ ∞

−∞
dz P (z), (4.51)

where P (x) is the scalar quantity constructed from K̄1(x, y) and K2(x, y) as

P (x) := K1(x, x)K̄2(x, x). (4.52)

(b) The potential pair (q, r) is recovered as

q(x) = −2K1(x, x) exp
(
− 2i

∫ ∞

x

dz P (z)
)
, (4.53)

r(x) = −2K̄2(x, x) exp
(
2i

∫ ∞

x

dz P (z)
)
. (4.54)

(c) The Jost solutions ψ(ζ, x) and ψ̄(ζ, x) to (1.1) are recovered as

ψ1(ζ, x) = ζ
(∫ ∞

x

dyK1(x, y)e
iζ2y

)
exp

(
− 2i

∫ ∞

x

dz P (z)
)
, (4.55)

ψ2(ζ, x) = eiζ
2x +

∫ ∞

x

dyK2(x, y)e
iζ2y, (4.56)

ψ̄1(ζ, x) =
(
e−iζ2x +

∫ ∞

x

dy K̄1(x, y)e
−iζ2y

)
exp

(
− 2i

∫ ∞

x

dz P (z)
)
, (4.57)

ψ̄2(ζ, x) = ζ

∫ ∞

x

dy K̄2(x, y)e
−iζ2y, (4.58)

where we recall that ψ1(ζ, x), ψ2(ζ, x), ψ̄1(ζ, x), and ψ̄2(ζ, x) are the components of the Jost
solutions defined in (2.1).

Proof. Using (4.43) and (4.46), we observe that the auxiliary scalar quantity P (x) defined in (4.52)
is expressed in terms of the potential pair (q, r) as

P (x) =
q(x) r(x)

4
. (4.59)

Hence, with the help of (1.11), (4.52), and (4.59) we obtain the first equality of (4.51). Using
(4.59) in (1.13), we obtain the second equality of (4.51). Thus, the proof of (a) is complete. Using
(4.51) in (4.43), we recover the potential q as in (4.53). Similarly, using (4.51) in (4.46), we recover
the potential r as in (4.54). Hence, the proof of (b) is also complete. We obtain the expressions
for the components of the Jost solutions ψ(ζ, x) and ψ̄(ζ, x) listed in (4.55)–(4.58), by using (4.51)
in (4.24)–(4.27), respectively. □

5. Solutions to the Chen-Lee-Liu system

In Section 4 we presented the solution to the inverse scattering problem for (1.6) by using the
Marchenko method. This has been done by using the data set {R(ζ), R̄(ζ), (A,B,C), (Ā, B̄, C̄)} as
input to the Marchenko system (4.39) and by recovering the potential pair (q, r) from the solution
K(x, y) to (4.39), as described in (4.53) and (4.54) with the quantity P (x) expressed as in (4.52).
If we use the time-evolved version of our input data set, via the Marchenko method we recover the
time-evolved potential pair (q, r). From the inverse scattering method [21, 24] we know that the
time-evolved potentials q(x, t) and r(x, t) form a solution to the integrable nonlinear system (1.1).
In this section, since we deal with the time-evolved quantities, we show the explicit t-dependence



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 19

in our notation for the potentials, the scattering coefficients, and the solution to the Marchenko
system.

In the next theorem, we provide the time evolution of the scattering coefficients for (1.6) and
the matrices C and C̄ appearing in the matrix triplets (A,B,C) and (Ā, B̄, C̄) used to describe
the bound-state information.

Theorem 5.1. Assume that the potentials q and r appearing in (1.6) belong to the Schwartz class
S(R) in x ∈ R for each fixed t ∈ R. Let the time evolution of the potentials q and r be governed
by the AKNS pair matrix T in (1.5). We have the following:

(a) The four reflection coefficients R(ζ, t), R̄(ζ, t), L(ζ, t), L̄(ζ, t) for (1.6) evolve in the time
variable t as

R(ζ, t) = R(ζ, 0) e4iλ
2t, R̄(ζ, t) = R̄(ζ, 0) e−4iλ2t, λ ∈ R, t ∈ R, (5.1)

L(ζ, t) = L(ζ, 0) e−4iλ2t, L̄(ζ, t) = L̄(ζ, 0) e4iλ
2t, λ ∈ R, t ∈ R,

where we recall that the parameter λ is related to the spectral parameter ζ as in (2.7).
(b) The four transmission coefficients Tl(ζ, t), Tr(ζ, t), T̄l(ζ, t), T̄r(ζ, t) do not change in time,

and hence we have

Tl(ζ, t) = Tl(ζ, 0), Tr(ζ, t) = Tr(ζ, 0), T̄l(ζ, t) = T̄l(ζ, 0), T̄r(ζ, t) = T̄r(ζ, 0).

Since the transmission coefficients do not change in time, for notational simplicity we use Tl(ζ),
Tr(ζ), T̄l(ζ), and T̄r(ζ) for Tl(ζ, t), Tr(ζ, t), T̄l(ζ, t), and T̄r(ζ, t), respectively.

(c) The four matrices A, Ā, B, B̄ appearing in (3.6) and (3.7) do not change in time. On the
other hand, the matrices C and C̄ appearing in (3.8) evolve in time as

C 7→ Ce4iA
2t, C̄ 7→ C̄e−4iĀ2t, (5.2)

where, for notational simplicity, we use C and C̄ to denote their values at t = 0.

Proof. The proof of (a)–(c) for the linear system (1.9) can be found in [13, Theorem 2.1], and the
corresponding proof for (1.6) is similarly obtained. □

With the help of (5.1), we obtain the time evolution of R̂(y, t) and ˆ̄R(y, t) defined in (4.2) as

R̂(y, t) =
1

2π

∫ ∞

−∞
dλ
R(ζ, 0)

ζ
e4iλ

2teiλy, ˆ̄R(y, t) =
1

2π

∫ ∞

−∞
dλ
R̄(ζ, 0)

ζ
e−4iλ2te−iλy. (5.3)

Next, using (5.2) and (5.3) in (4.37) we obtain the time-evolved Marchenko kernels Ω(y, t) and
Ω̄(y, t) as

Ω(y, t) = R̂(y, t) + Ce4iA
2teiAyB, Ω̄(y, t) = ˆ̄R(y, t) + C̄e−4iĀ2te−iĀyB̄. (5.4)

We remark on the symmetry and elegance in the connection between (5.3) and (5.4). The scalar

term e4iλ
2teiλy appearing in the integrand in the first equality of (5.3) suggests the appearance

of the matrix exponential e4iA
2teiAy in the first equality of (5.4). When there is a simple bound

state at λ = λj of multiplicity 1, the eigenvalue of A and the matrix A coincide. Thus, the

scalar λ appearing in e4iλ
2teiλy in the first equality of (5.3) is naturally replaced by the matrix

A in e4iA
2teiAy in the first equality of (5.4). In a similar manner, the scalar term e−4iλ2te−iλy

appearing in the integrand in the second equality of (5.3) suggests the appearance of the matrix

exponential e−4iĀ2te−iĀy in the second equality of (5.4). When there is a simple bound state
at λ = λ̄j of multiplicity 1, the eigenvalue of Ā and the matrix Ā coincide. Thus, the scalar λ

appearing in e−4iλ2te−iλy in the second equality of (5.3) is naturally replaced by the matrix Ā in

e−4iĀ2te−iĀy.
In the reflectionless case, i.e. when the reflection coefficients for (1.6) are all zero, there are

various restrictions on the bound-state information presented in (3.1). Hence, in the reflectionless
case, there are restrictions on the matrix triplets (A,B,C) and (Ā, B̄, C̄) appearing in (3.6)–(3.8).
We refer the reader to Theorem 4.3 of [13] for the restrictions on the bound-state information
for (1.9) in the reflectionless case. Since the transmission coefficients for (1.6) and for (1.9) are
related to each other as in (2.39) and (2.40), the restrictions on the bound-state information for



20 M. UNLU EJDE-2025/97

(1.6) in the reflectionless case can be obtained from the corresponding restrictions for (1.9). For
example, in the reflectionless case, we must have N = N̄ , where N and N̄ are the nonnegative
integer quantities defined in (3.9). In the reflectionless case, with the help of (4.40) of [13] and
(2.40), we obtain the restriction that the transmission coefficients Tr(ζ) and T̄r(ζ) for (1.6) have
to be related to each other as

Tr(ζ)T̄r(ζ) = e−iµ/2, λ ∈ R, (5.5)

where we recall that λ and ζ are related to each other as in (2.7) and µ is the constant appearing
in (1.13). In that case, the transmission coefficients Tr(ζ) and T̄r(ζ) each have meromorphic
extensions to the entire complex λ-plane, and hence the bound-state λj-values and λ̄j-values
become restricted. Such restrictions must be taken into account in considering the analysis of
solutions to the linear system (1.6) and the nonlinear system (1.1) in the reflectionless case.

In the reflectionless case, from (5.2) we obtain

Ω(y, t) = Ce4iA
2teiAyB, Ω̄(y, t) = C̄e−4iĀ2te−iĀyB̄. (5.6)

In the next theorem, in the reflectionless case without imposing any of the restrictions on the
matrix triplets, we obtain the solution to the Marchenko system (4.39) in terms of the matrix
triplets (A,B,C) and (Ā, B̄, C̄) with the help of the input {Ω(y, t), Ω̄(y, t)} given in (5.4).

Theorem 5.2. Assume that the potential pair (q, r) appearing in (1.6) belongs to the Schwartz
class S(R) in x ∈ R for each fixed t ∈ R, where the time evolution of the potential pair is governed
by the AKNS pair matrix T given in (1.5). Then, in the reflectionless case, the solution K(x, y, t)
to the Marchenko system (4.39) is

K1(x, y, t) = −C̄e−iĀxΓ̄(x, t)−1e−iĀy−4iĀ2tB̄, (5.7)

K̄1(x, y, t) = C̄e−iĀxΓ̄(x, t)−1e−iĀx−4iĀ2tM̄AeiA(x+y)+4iA2tB, (5.8)

K2(x, y, t) = CeiAx Γ(x, t)−1eiAx+4iA2tMĀe−iĀ(x+y)−4iĀ2tB̄, (5.9)

K̄2(x, y, t) = −CeiAx Γ(x, t)−1eiAy+4iA2tB, (5.10)

where K1(x, y, t), K2(x, y, t), K̄1(x, y, t), K̄2(x, y, t) are the entries of K(x, y, t) appearing in
(4.13). The 2× 2 matrix-valued functions Γ(x, t) and Γ̄(x, t) appearing (5.7)–(5.10) are expressed
in terms of the matrix triplets (A,B,C) and (Ā, B̄, C̄) as

Γ(x, t) := I − eiAx+4iA2tMĀe−2iĀx−4iĀ2tM̄eiAx, (5.11)

Γ̄(x, t) := I − e−iĀx−4iĀ2tM̄Ae2iAx+4iA2tMe−iĀx, (5.12)

with I denoting the 2×2 identity matrix and the quantities M and M̄ being the 2×2 matrix-valued
constants expressed in terms of the matrix triplets (A,B,C) and (Ā, B̄, C̄) as

M :=

∫ ∞

0

dzeiAz BC̄e−iĀz, M̄ :=

∫ ∞

0

dze−iĀzB̄ CeiAz. (5.13)

Proof. From (5.6), by taking the y-derivatives we obtain

Ω′(y, t) = iCAe4iA
2teiAyB, Ω̄′(y, t) = −iC̄Āe−4iĀ2te−iĀyB̄, (5.14)

where the prime is used to denote the y-derivative. Using (5.6) and (5.14) in the uncoupled system
we recover K1(x, y, t) and K̄2(x, y, t) listed in (5.7) and (5.10), respectively. We then use (5.7) in
the integrand in the first line of (4.42) and use (5.10) in the integrand in the second line of (4.42)
and recover K̄1(x, y, t) and K2(x, y, t) given in (5.8) and (5.9), respectively. Since the process is
similar to the proof of [12, Theorem 5.4], we omit the details. We remark that since the eigenvalues
of A are all located in the upper-half complex λ-plane and the eigenvalues of Ā are all located in
the lower-half complex λ-plane, the two integrals in (5.13) both exist. □

In the next theorem, we recover the potential pair (q, r) appearing in (1.6) in terms of the
matrix triplets (A,B,C) and (Ā, B̄, C̄) in the reflectionless case.



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 21

Theorem 5.3. Assume that the potential pair (q, r) appearing in (1.6) belong to the Schwartz
class S(R) in x ∈ R for each fixed t ∈ R, where the time evolution of the potential pair is governed
by the matrix T in (1.5). Let the quantities Ω(y, t) and Ω̄(y, t) be the time-evolved reflectionless
Marchenko kernels defined in (5.6). Then, we have the following:

(a) The corresponding key quantity E(x, t) defined in (1.11) can explicitly be written in terms
of the matrix triplets (A,B,C) and (Ā, B̄, C̄) as

E(x, t) = exp
(
2i

∫ x

−∞
dz P (z, t)

)
, (5.15)

where P (x, t) is the scalar-valued function of x and t defined in (4.52) with K̄1(x, x, t) and
K2(x, x, t) there explicitly expressed in terms of the matrix triplets as

K1(x, x, t) = −C̄e−iĀxΓ̄(x, t)−1e−iĀx−4iĀ2tB̄, (5.16)

K̄2(x, x, t) = −CeiAx Γ(x, t)−1eiAx+4iA2tB. (5.17)

The 2 × 2 matrix-valued quantity Γ(x, t) in (5.11) is explicitly determined by the matrix triplets
(A,B,C) and (Ā, B̄, C̄) as described in (5.16). Similarly, the 2× 2 matrix-valued quantity Γ̄(x, t)
in (5.12) is explicitly determined by the matrix triplets (A,B,C) and (Ā, B̄, C̄) as described in
(5.17).

(b) The corresponding potential pair (q, r) appearing in the linear system (1.6) can explicitly be
expressed in terms of the matrix triplets (A,B,C) and (Ā, B̄, C̄) as

q(x, t) =
(
2C̄e−iĀxΓ̄(x, t)−1e−iĀx−4iĀ2tB̄

)
exp

(
− 2i

∫ ∞

x

dz P (z, t)
)
, (5.18)

r(x, t) =
(
2CeiAx Γ(x, t)−1eiAx+4iA2tB

)
exp

(
2i

∫ ∞

x

dz P (z, t)
)
. (5.19)

Proof. We get (5.16) and (5.17) from (5.7) and (5.10), respectively, by letting y → x+ there. Thus,
we obtain (5.15) from the first equality of (4.51) with the help of (4.52) by using (5.16) and (5.17)
on the right-hand side of (4.52). This completes the proof of (a). We obtain (5.18) from (4.53) by
using (5.16) on the right-hand side of (4.53). Similarly, we obtain (5.19) by using (5.17) on the
right-hand side of (4.54). □

6. Explicit examples

In this section, we illustrate the solution method for the Chen-Lee-Liu system (1.1) developed
in Section 5 with two explicitly solved examples.

In the first example below, we construct a one-soliton solution with multiplicity 1 as a solution
to the Chen-Lee-Liu system (1.1). This is done by solving the inverse scattering problem for
(1.6) in the reflectionless case via the Marchenko method developed in Section 4. As input to the
Marchenko system (4.39) we use a specific pair of matrix triplets, where we choose the size of each
of the six matrices in the triplets as 1 × 1. For notational simplicity in our example, we write a
1× 1 matrix as a scalar quantity.

Example 6.1. In the reflectionless case, we use the matrix triplets (A,B,C) and (Ā, B̄, C̄) given
by

A =
[
i
]
, B =

[
1
]
, C =

[
i
]
, Ā =

[
−i

]
, B̄ =

[
1
]
, C̄ =

[
−i

]
. (6.1)

Using (6.1) in (5.13), we construct the 1× 1 constant matrices M and M̄ as

M = − i

2
, M̄ =

i

2
. (6.2)

Next, using (6.1) and (6.2) in (5.11) and (5.12), we obtain the 1× 1 matrices Γ(x, t) and Γ̄(x, t) as

Γ(x, t) = 1 +
ie−4x

4
, Γ̄(x) = 1− ie−4x

4
. (6.3)



22 M. UNLU EJDE-2025/97

Then, we use (6.1) and (6.3) in (5.16) and (5.17) and construct the scalar quantities K1(x, x, t)
and K̄2(x, x, t), respectively, and we obtain

K1(x, x, t) = − 4e2x+4it

1 + 4ie4x
, K̄2(x, x, t) = − 4e2x−4it

1− 4ie4x
. (6.4)

Next, using (6.4) in (4.52) we obtain the quantity P (x, t) as

P (x, t) =
16e4x

1 + 16e8x
. (6.5)

Finally, we use (6.4) and (6.5) in (4.51), (4.53), (4.54), and we recover the quantity E(x, t), the
constant µ, and the potentials q(x, t) and r(x, t) as

E(x, t) = e2i tan−1(4e4x), µ = 2π, (6.6)

q(x, t) =
8e2x+4it+2i tan−1(4e4x)

1 + 4ie4x
, r(x, t) =

8e2x−4it−2i tan−1(4e4x)

1− 4ie4x
, x ∈ R, t ∈ R. (6.7)

From (6.7) we observe that the potentials q(x, t) and r(x, t) satisfy

r(x, t) = q(x, t)∗,

where we use an asterisk to denote complex conjugation. From (6.7) we also see that the potentials
q(x, t) and r(x, t) have no singularities and they belong to the Schwartz class S(R) in x for each
fixed t ∈ R. Since the 1× 1 matrices A and Ā in the input data set (6.1) correspond to the poles
of Tr(ζ) and T̄r(ζ) in the upper-half and lower-half complex λ-planes, respectively, with the help
of (5.5) we determine those two right transmission coefficients as

Tr(ζ) = −λ+ i

λ− i
, T̄r(ζ) =

λ− i

λ+ i
, λ ∈ C,

where the value of µ appearing in the second equality in (6.6) is taken into account and we recall
that the parameter λ is related to the spectral parameter ζ as in (2.7). In this example, as seen
from the first equality of (6.6), the quantity E(x, t) is independent of t even though the potentials
q(x, t) and r(x, t) given in (6.7) contain the parameter t. On the other hand, the quantities |q(x, t)|
and |r(x, t)| do not change in t. Since q(x, t) and r(x, t) are complex valued, in Figure 1 we have
plotted |q(x, t)| and |r(x, t)| as functions of x. From (6.7) we have |r(x, t)| = |q(x, t)| for all x ∈ R
and t ∈ R. Hence, in this example, the soliton represented by |q(x, t)| or |r(x, t)| does not move
in time.

-6 -4 -2 0 2 4

1

2

3

4
q(x, t)

-6 -4 -2 0 2 4

1

2

3

4
r(x, t)

Figure 1. Snapshots for |q(x, t)| and |r(x, t)| in Example 6.1 corresponding to
the input data set in (6.1).

In the next example, we illustrate a double-pole soliton solution to (1.1). As in Example 6.1
this is done by solving the inverse scattering problem for (1.6) in the reflectionless case via the
Marchenko method. As input to the Marchenko system (4.39), we use a pair of matrix triplets
with the 2× 2 matrices A and Ā presented in their Jordan canonical forms.



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 23

Example 6.2. In the reflectionless case, we choose the input data set to the Marchenko system
as

A =

[
i 1
0 i

]
, B =

[
0
1

]
, C =

[
i 1

]
, Ā =

[
−i 1
0 −i

]
, B̄ =

[
0
1

]
, C̄ =

[
−i 1

]
. (6.8)

We recover the potential pair (q, r) in terms of the matrix triplets (A,B,C) and (Ā, B̄, C̄) as
follows. Using (6.8) in (5.13), we construct the 2× 2 constant matrices M and M̄ as

M =

[
1
4 0

− i
2

1
4

]
, M̄ =

[
1
4 0
i
2

1
4

]
. (6.9)

Then, using (6.8) and (6.9) in (5.11) and (5.12), we obtain the 2× 2 matrices Γ(x, t) and Γ̄(x, t),
respectively, as

Γ(x) =

[
Γ11 Γ12

Γ21 Γ22

]
, Γ̄(x) =

[
Γ∗
11 Γ∗

12

Γ∗
21 Γ∗

22

]
, (6.10)

where we have defined

Γ11 := 1 +
e−4x

16

(
−i+ 8

(
ix2 − 4xt+ 6t+ 32it2

))
,

Γ12 :=
e−4x

16

(
−1 + 4x2 − 8x3 − 128t2(−1 + 2x)− 32it(x− 12)

)
,

Γ21 :=
e−4x

4
(−1 + 2x− 8it) ,

Γ22 := 1 +
e−4x

16

(
i(3− 8x+ 8x2) + 16t(−1 + 2x)

)
,

and we recall that an asterisk denotes complex conjugation. Having constructed the matrices
Γ(x, t) and Γ̄(x, t), we use (6.8) and (6.10) in (5.16) and (5.17), and we obtain K1(x, x, t) and
K̄2(x, x, t), respectively, as

K1(x, x, t) =
32e2x+4it

(
−x+ 8e4x (2ix− i+ 8t)− 4it

)
32e4x (1− 4x+ 8x2 + 128t2 + 16it)− i+ 256ie8x

, (6.11)

K̄2(x, x, t) =
32e2x−4it

(
−x+ 8e4x (−2ix+ i+ 8t) + 4it

)
32e4x (1− 4x+ 8x2 + 128t2 − 16it) + i− 256ie8x

. (6.12)

Next, we use (6.11) and (6.12) in (4.52), and we obtain the quantity P (x, t) as

P (x, t) =
1024e4x w1 w

∗
1

w2 w∗
2

, (6.13)

where we have defined

w1 := ix− 4 t− 8ie4x(−i+ 8t+ 2i x),

w2 := −1 + 256e8x − 32ie4x
(
1− 4x+ 8x2 + 16it+ 128t2

)
.

From (6.13) we observe that the quantity P (x, t) is real valued for all x ∈ R and t ∈ R. Finally,
using (6.11), (6.12), and (6.13) in (4.51), (4.53), and (4.54), we recover the quantity E(x, t), the
constant µ, and the potentials q(x, t) and r(x, t) as

E(x, t) = exp
(
− 2i tan−1(w3/w4)

)
, µ = 4π, (6.14)

q(x, t) = −64w1

w2
exp

(
2x+ 4it− 2i tan−1(w3/w4)

)
, x ∈ R, t ∈ R, (6.15)

r(x, t) = q(x, t)∗, x ∈ R, t ∈ R, (6.16)

where we have defined

w3 := 32e4x
(
1− 4x+ 8x2 + 128t2

)
, w4 := −1 + 256e8x + 512e4xt.

From (6.15) and (6.16) we observe that the potentials q(x, t) and r(x, t) satisfy |r(x, t)| = |q(x, t)|
and they each belong to the Schwartz class S(R) in x for t ∈ R. We also see from (6.14) that the
quantity E(x, t) depends on t, whereas in Example 6.1 the quantity E(x, t) given in (6.6) does



24 M. UNLU EJDE-2025/97

not depend on t. The eigenvalues of the 2× 2 matrices A and Ā correspond to the poles of Tr(ζ)
and T̄r(ζ) in the upper-half and lower-half complex λ-planes, respectively. Hence, with the help
of (5.5) we see that those two right transmission coefficients are given by

Tr(ζ) =
(λ+ i

λ− i

)2

, T̄r(ζ) =
(λ− i

λ+ i

)2

, λ ∈ C, (6.17)

where the value of µ listed in the second equality of (6.14) is taken into account and we recall
that the parameters λ and ζ are related to each other as in (2.7). As seen from (6.17), each
transmission coefficient has a double pole. Since q(x, t) is complex valued and |r(x, t)| = |q(x, t)|,
we only discuss the time evolution of |q(x, t)|. In Figure 2 we show the snapshots for |q(x, t)| at
t = −5, t = −0.2, t = 0, t = 0.1, t = 0.2, and t = 5, respectively. As seen from Figure 2, there
are two solitons that are initially far apart. They move toward each other and a Mathematica
animation shows that their speeds increase as they get closer. Then, the two solitons interact with
each other nonlinearly, and then they move away from each other. As they move away from each
other, they regain their individual shapes.

-6 -4 -2 0 2 4

1

2

3

4
q(x, -5)

-6 -4 -2 0 2 4

1

2

3

4
q(x, -0.2)

-6 -4 -2 0 2 4

1

2

3

4
q(x, 0)

-6 -4 -2 0 2 4

1

2

3

4
q(x, 0.1)

-6 -4 -2 0 2 4

1

2

3

4
q(x, 0.2)

-6 -4 -2 0 2 4

1

2

3

4
q(x, 5)

Figure 2. Snapshots for |q(x, t)| in Example 6.2 at t = −5, t = −0.2, t = 0,
t = 0.1, t = 0.2, and t = 5, respectively.

In the reflectionless case, we have prepared a Mathematica notebook that allows the user
to input the entries of the two matrix triplets corresponding to any number of bound states
with any multiplicities. By using the method of Section 5, our Mathematica notebook evaluates
all the relevant quantities and yields the output including the scalar quantity E(x, t) in (1.9),
the constant µ in (1.11), the potentials q(x, t) and r(x, t) appearing in (1.1) and (1.6), and the
transmission coefficients Tr(ζ) and T̄r(ζ) appearing in (5.5). Our Mathematica notebook also
allows the user to animate |q(x, t)| and |r(x, t)| to observe the time evolution of soliton solutions
to (1.1) corresponding to any number of bound states and any number of multiplicities.

The advantage of using matrix exponentials in expressing explicit solutions to (1.1) becomes
clear as the number of bound states or their multiplicities become large. As seen from (5.18)
and (5.19), the potentials q(x, t) and r(x, t) are expressed in a compact form with the help of
matrix exponentials constructed by using a pair of matrix triplets. By “unpacking” those matrix
exponentials, we can express q(x, t) and r(x, t) in terms of elementary functions, where those
latter expressions become extremely lengthy as the number of bound states and their multiplicities
increase, whereas the compact expressions in (5.18) and (5.19) involving matrix exponentials and
matrix triplets remain unchanged no matter how many bound states we have and no matter how
large their multiplicities are.



EJDE-2025/97 DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS 25

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26 M. UNLU EJDE-2025/97

Mehmet Unlu

Department of Mathematics, Recep Tayyip Erdogan University, 53100 Rize, Turkey

Email address: mehmet.unlu@erdogan.edu.tr


	1. Introduction
	2. The direct scattering problem
	3. Bound states
	4. The Marchenko method
	5. Solutions to the Chen-Lee-Liu system
	6. Explicit examples
	References