Adv Syst Sci Appl 2020; 04:125–131
Published online at https://ijassa.ipu.ru.

On Some Global Properties of Multivalued Simple Waves

Dmitry V. Tunitsky1

1V.A. Trapeznikov Institute of Control Sciences of RAS, Profsoyuznaya 65, 117342 Moscow, Russia

Abstract: The Cauchy problem for one-dimensional quasilinear wave equation is considered.
In the case that its solutions are multivalued simple waves, we derive explicit expressions in
quadratures by introducing global characteristic coordinates. The main result of the paper is a
theorem on some global properties of multivalued simple waves.

Keywords: hyperbolic quasilinear wave equation; multivalued solution; Cauchy problem;
simple wave

INTRODUCTION

A number of models in continuum mechanics concern the wave equation

zxx − g2(zy)zyy = 0, (0.1)

where g = g(q) is a given positive coefficient,

g(q) > 0, (0.2)

cf. [1], [2], and [3], and the Monge notations

p = zx(x, y), q = zy(x, y) (0.3)

are used. The variable x designates time, y is the spatial coordinate, and z is the displacement
of continuum. In particular, vibration of a string, cf. [3], [1]; wave propagation in a bar
of elastic-plastic material, cf. [4]; and isentropic flows of a compressible gas with plane
symmetry, cf. [4], [5], are described by equations of the type (0.1).

Equation (0.1) is a quasilinear second-order partial differential equation with two
independent variables x and y and unknown function

z = z(x, y). (0.4)

Due to condition (0.2) it is hyperbolic.
It is common knowledge what is a classical solution of equation (0.1). Classical solutions

have a rather serious drawback: in finite time singularities, a so called gradient catastrophes,
can develop in them, cf. [2] and [1]. The latter means that there is a point (x, y) such that
in some its vicinity a classical solution z (0.4) itself and its first derivatives p and q (0.3) are
bounded but at least one of second derivatives is unbounded. This is a motivation and reason
to generalize a notion of classical solution and define the notion of a multivalued solution,
which is a relatively well-known one, cf. [6], [7], [8], and [9].

∗Corresponding author: dtunitsky@yahoo.com



126 D.V. TUNITSKY

1. MULTIVALUED SOLUTIONS

The differential 2-form
ω2 = dp ∧ dy − g(q)dx ∧ dq, (1.5)

is in an obvious way associated with the left hand side of the equation (0.1), cf. [8], and the
linear differential form and its exterior derivative

ω0 = dz − pdx− qdy,
ω1 = dω0 = dx ∧ dp+ dy ∧ dq (1.6)

are associated with equalities (0.3). An immersion

σ : S −→ R5 (1.7)

of a two-dimensional Hausdorff paracompact manifold S is a multivalued solution of
equation (0.1) if it satisfies the following system of exterior differential equations

σ∗ω0 = 0, σ∗ω1 = 0, σ∗ω2 = 0, (1.8)

cf. [7] and [8]. Obviously, the graphic of a classical solution is a multivalued solution
but not vice versa. And it can be proven that any classical solution is a part of maximal
multivalued solution, cf. [10]. If a gradient catastrophe takes place at a point s ∈ S,
then (dσ∗x ∧ dσ∗y) (s) = 0. Multivalued solutions have a serious advantage over classical
solutions because gradient catastrophes do not happen to them; cf. [10].

The linear differential forms

ωj,1 = dp− (−1)jg(q)dq,

ωj,2 = dy − (−1)jg(q)dx
(1.9)

will be called characteristic. It is not hard to see that the equalities

ω2 − g(p)ω1 = ω1,1 ∧ ω1,2,

ω2 + g(p)ω1 = ω2,1 ∧ ω2,2,

ω2 =
1

2
(ω1,1 ∧ ω1,2 + ω2,1 ∧ ω2,2),

ω1 =
1

2g(p)
(ω1,1 ∧ ω1,2 − ω2,1 ∧ ω2,2)

are through for characteristic forms ωj,1, ωj,2 (1.9) and 2-forms ω2 (1.5) and ω1 (1.6). Thence,
an immersion σ (1.7) is a multivalued solution of the equation (0.1) iff

σ∗ω0 = 0,

σ∗(ω1,1 ∧ ω1,2) = 0,

σ∗(ω2,1 ∧ ω2,2) = 0.

(1.10)

A curve
γ : Γ −→ R5,

where Γ is a one-dimensional connected manifold, will be called a characteristic curve of the
equation (0.1) that belongs to the j-th family, j = 1, 2, if

γ∗ω0 = 0, γ∗ωj,1 = 0, γ∗ωj,2 = 0. (1.11)

Suppose an immersion σ (1.7) is a multivalued solution of equation (0.1). Then equations
(1.10) are through for this immersion. Hence the pull-backs σ∗ωj,1 and σ∗ωj,2 of the

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MULTIVALUED SIMPLE WAVES 127

characteristic forms ωj,1 and ωj,1 (1.9) are linearly dependent and the linear algebraic
equations

σ∗ωj,1 = 0, σ∗ωj,2 = 0 (1.12)

uniquely define the one-dimensional subbundle of the tangent bundle TS for j = 1, 2.
Therefore, by the Frobenius theorem [11], for any point s ∈ S and number j = 1, 2 there
exists the maximal integral manifold

γj,s : Γj,s −→ S, (1.13)

of the system
γ∗j,sωj,1 = 0, γ∗j,sωj,2 = 0, (1.14)

containing the point s, where Γj,s is a connected one-dimensional manifold.
It follows from the equations (1.11), (1.12), and (1.14) that the composition σ ◦ γj,s of

the maps (1.7) and (1.13) is a characteristic curve of the equation (0.1), belonging to the j-th
family. Of cause, for classical solutions conventional characteristic curves are obtained this
way.

2. CAUCHY PROBLEM

An initial curve or initial value for equation (0.1) is a smooth curve

l : (−∞,+∞) −→ R5 (2.15)

that satisfies the conditions

ω0(l̇(τ)) = 0,

ω2
1,1(l̇(τ)) + ω2

1,2(l̇(τ)) 6= 0,

ω2
2,1(l̇(τ)) + ω2

2,2(l̇(τ)) 6= 0

(2.16)

for −∞ < τ < +∞. The last two conditions mean that the initial curve l (2.15) is not
characteristic, i.e. it is free, see (1.11).

A classical-type initial value problem at x = 0 for equation (0.1) is defined by conditions

z(0, y) = z0(y), zx(0, y) = q0(y), (2.17)

where
z0 : (−∞,+∞) −→ R,
q0 : (−∞,+∞) −→ R (2.18)

are given functions. This initial values determine initial curve l (2.15) with the coordinates

l∗x(τ) = 0,

l∗y(τ) = τ,

l∗p(τ) = p0(τ),

l∗q(τ) = q0(τ),

l∗z(τ) = z0(τ).

(2.19)

Obviously, this curve is an immersion and meets conditions (2.16) since

ω2
1,1(l̇(τ)) = ω2

2,2(l̇(τ)) = 1

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128 D.V. TUNITSKY

according to expressions (1.9).
If for a multivalued solution (1.7) of equation (0.1) there exists an imbedding

L : (−∞,+∞) −→ S (2.20)

such that
l = σ ◦ L, (2.21)

then σ is called a solution of Cauchy problem (0.1), (2.15) and L – an initial embedding.
Thus a solution of the Cauchy problem (0.1), (2.15) is a pair (σ, L) of an immersion σ
(1.7), satisfying the system (1.10), and imbedding L (2.20), satisfying (2.21). A multivalued
solution (σ, L) of the initial value problem (0.1), (2.15) is said to be definite if for any point
s ∈ S and number j = 1, 2 the intersection γj,s(Γj,s) ∩ L(−∞,+∞) of the image of initial
imbedding L (2.20) and characteristic curve γj,s (1.13) consists of exactly one point.
Theorem 2.1:
(Characteristic uniformization, [10]) Let (σ, L) be a definite solution (1.7), (2.20) of the
Cauchy problem (0.1), (2.15)–(2.16). Then there exists a unique diffeomorphism

Φ : S −→ Φ(S) (2.22)

such that the following three properties hold.
(a) The image Φ(S) is a subset of R2 and contains its diagonal

δ = {(τ, τ) ∈ R2| −∞ < τ < +∞}.

(b) For −∞ < τ < +∞ the initial imbedding L (2.20) satisfies the equality

Φ ◦ L(τ) = (τ, τ).

(c) Images Φ ◦ γj,s(τ) of characteristic curves γj,s (1.13), where s ∈ S, lie in coordinate
straight lines u = const if j = 1 and v = const if j = 2 for τ ∈ Γj,s.

A diffeomorphism Φ (2.22), which is uniquely defined by Theorem 2.1, is called a
characteristic uniformization of the definite solution (σ, L). The coordinate plane of the
parameters u, v and the same coordinates in it are called characteristic as well. The image
Φ(S) of the uniformization Φ (2.22) is a uniformized domain and the composition

ψ = σ ◦ Φ−1 (2.23)

is a uniformized solution.

3. SIMPLE WAVES

A solution σ (1.7) is said to be a simple wave, if the equality

dσ∗p ∧ dσ∗q = 0 (3.24)

holds for it, cf. [4]. It is possible to show that a definite solution (σ, L) of the Cauchy problem
(0.1), (2.17) is a simple wave iff either

p0(y) = −G(z′0(y)), (3.25)

or
p0(y) = G(z′0(y)), (3.26)

where G = G(q) is a primitive function of g = g(q).

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MULTIVALUED SIMPLE WAVES 129

By definition (3.24), in case of simple waves linearization of equation (0.1) by means
of hodograph transformation is not applicable. But it is possible to apply Theorem 2.1 and
use characteristic uniformization Φ (2.22) in this case instead. Indeed, by Theorem 2.1,
definition (1.13)–(1.14) of characteristic curve, and initial values (3.25) and (3.26) we get
for coordinates x, y, p, q, and z of uniformized solution (2.23) the following equations

∂

∂v
y ψ∗ω1,1 = pv + g(p)qv = 0,

∂

∂v
y ψ∗ω1,2 = yv + g(p)xv = 0,

∂

∂v
y ψ∗ω0 = zv − pxv − qyv,

∂

∂u
y ψ∗ω2,1 = pu − g(q)qu = 0,

∂

∂u
y ψ∗ω2,2 = yu − g(p)xu = 0,

∂

∂u
y ψ∗ω0 = zu − pxu − qyu

(3.27)

where the symbol y denotes the interior multiplication, see [11] (section 2.11), and the initial
values

z0(τ, τ) = z0(τ),

q0(τ, τ) = z′0(τ),

p0(τ, τ) = −G(z′0(τ))

(3.28)

in the case of (3.25) and
z0(τ, τ) = z0(τ)),

q0(τ, τ) = z′0(τ)),

p0(τ, τ) = G(z′0(τ))

(3.29)

in the case of (3.26).
Integration of Cauchy problems (3.27), (3.28) and (3.27), (3.29) allows to represent

characteristic uniformization (2.23) of a simple wave in quadratures.

Proposition 3.1:
In characteristic coordinates u and v the coordinates x, y, p, q, and z of uniformization
(2.23) of any definite multivalued solution (σ, L) of the Cauchy problem (0.1), (2.17) that is
a maximal simple wave are representated in the following way:

x(u, v) =
h(u, v)

2
√
g0(v)

,

y(u, v) = v +

√
g0(v)

2
h(u, v),

z(u, v) = z0(v) +
p0(v) + g0(v)z′0(v)

2
√
g0(v)

h(u, v),

p(u, v) = p0(v),

q(u, v) = z′0(v)

(3.30)

Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020)



130 D.V. TUNITSKY

in case (3.25) and

x(u, v) =
h(u, v)

2
√
g0(u)

,

y(u, v) = u−
√
g0(u)

2
h(u, v),

z(u, v) = z0(u) +
p0(u) + g0(u)z′0(u)

2
√
g0(u)

h(u, v),

p(u, v) = p0(u),

q(u, v) = z′0(u)

(3.31)

in case (3.26). Here (u, v) ∈ R2,

g0(τ) = g(z′0(τ)), h(u, v) =

u∫
v

dτ√
g0(τ)

.

Put
π : R5 3 (x, y, p, q, z) 7−→ (x, y) ∈ R2.

The following statement is an immediate corollary of representations (3.30) and (3.31) from
proposition 3.1. It gives sufficient conditions for projection π ◦ σ of a maximal simple wave
(σ, L) to be a proper map of degree 1.
Theorem 3.1:
If

lim
τ→∞

g0(τ)

τ
= 0, lim

τ→∞
|h(τ, 0)| = +∞,

then the projection π ◦ σ of the maximal simple wave (σ, L) is a proper map of degree 1 and,
therefore,

π ◦ σ(R2) = R2.

ACKNOWLEDGEMENTS

The reported study was funded by RFBR and JSPS (project number 19- 51-50005) and
RFBR (project number 20-01-00610).

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MULTIVALUED SIMPLE WAVES 131

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	Multivalued solutions
	Cauchy problem
	Simple waves