Adv Syst Sci Appl 2025; 02:51–57
Published online at https://ijassa.ipu.ru.

Multivalued Solutions of the Cauchy Problem
for One Quasilinear System

Elena N. Kushner

Bauman Moscow State Technical University, Moscow, Russia

Abstract: The article proposes a generalization of the method of characteristics to quasilinear
systems of evolutionary equations with one spatial variable. Such systems are a special case of
hydrodynamic type systems. It is shown that in some cases self-intersection of solution graphs is
possible. An example of a system of two equations is considered in detail.

Keywords: Hydrodynamic type systems, multivalued solutions, jet spaces, caustics, self-
intersection, Cauchy problem

1. INTRODUCTION

Consider the following first order quasilinear system of m equations
∂u1

∂t
+ A(u1, . . . , um)

∂u1

∂x
= 0,

. . . . . . . . . . . . . . . . . . . . . . . . . . .

∂um

∂t
+ A(u1, . . . , um)

∂um

∂x
= 0.

(1.1)

Here t is time, x is a spatial coordinate, the function A ∈ C∞(Rm). This is a special case of
a hydrodynamic type systems and a generalization of the scalar Euler–Hopf equation

ut + uux = 0. (1.2)
It is well known that solutions of equation (1.2) become discontinuous at some moment of
time even for a smooth initial condition. A similar phenomenon occurs for the considered
system. In addition, we will show that for such systems the effect of self-intersection of the
graphs of their solutions is possible.

Consider the following Cauchy problem for equation (1.1)

ui|t=0 = U i(x), i = 1, . . . ,m. (1.3)

Assume that the initial functions U i belong to the class C∞.

2. MULTIVALUED SOLUTIONS

Let J1 = J1(2,m) be the 1-jet space with canonical coordinates (see [1, 2])

t, x, u1
0,0, u

1
1,0, u

1
0,1, . . . , u

m
0,0, u

m
1,0, u

m
0,1; i = 1, . . . ,m.

∗Corresponding author: ekushner@ro.ru



52 E. KUSHNER

The Cartan distribution C is generated by the differential 1-forms

κi = dui
0,0 − ui

1,0dt− ui
0,1dx,

i.e.

C(θ) =
m⋂
i=1

kerκi |θ ⊂ TθJ
1.

System (1.1) generates a 2(m+ 1)-dimensional submanifold

E =
{
ui
1,0 + Aui

0,1 = 0
∣∣ i = 1, . . . ,m

}
⊂ J1.

A two-dimensional integral manifold of the Cartan distribution lying in E is called a
multivalued solution of system (1.1).

Let us show a method for constructing multivalued solutions to Cauchy problem (1.3).
Define the following vector field on the 0-jet space J0(2,m):

X = − ∂

∂t
− A(u1

0,0, . . . , u
m
0,0)

∂

∂x
.

Its prolongation to J1(2,m) is

X(1) = X +
m∑
j=1

(
m∑
i=1

∂A

∂ui
0,0

ui
1,0

)
uj
0,1

∂

∂uj
1,0

+
m∑
j=1

(
m∑
i=1

∂A

∂ui
0,0

ui
0,1

)
uj
0,1

∂

∂uj
0,1

.

Lemma 2.1:
The vector field X(1) is tangent to E .

Proof
The Lie derivative of the function Fk = uk

1,0 + Auk
0,1 is

X(1)(Fk) =

(
m∑
i=1

∂A

∂ui
0,0

ui
1,0

)
uk
0,1 +

(
m∑
i=1

∂A

∂ui
0,0

ui
0,1

)
Auk

0,1.

Therefore, the restriction
X(1)(Fk)|E = 0

for k = 1, . . . ,m.

3. CAUCHY DATA

The initial data (1.3) define a curve in the 1-jet space:

K :

{
t = 0, x = χ, ui

0,0 = U i(χ), ui
0,1 =

dU i

dχ
, ui

1,0 = −Ã
dU i

dχ

∣∣∣∣ i = 1, . . . ,m

}
,

where à = A(U1(χ)), . . . , Um(χ)). This curve is called Cauchy curve.
Let Φτ be a flow of X(1). Since the vector field X(1) is transversal to the hyperplane t = 0,

we see that X(1)
a /∈ TaK for any point , i.e. trajectories of the vector field intersect the Cauchy

curve. So, for sufficiently small τ , the union of Φτ (K) forms a surface L:

L =
⋃
τ

Φτ (K). (3.4)

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



MULTIVALUED SOLUTIONS OF THE CAUCHY PROBLEM... 53

Theorem 3.1:
Surface (3.4) is a multivalued solution of equation (1.1).

Proof
Due to Lemma 2.1, the vector field X(1) is tangent to E . Then L ⊂ E . It remains to prove that
L is an integral manifold of the Cartan distribution.

The Cauchy curve is an integral curve of the Cartan distribution. Since the vector field
X(1) is contact, the curve Φτ (K) is an integral for the Cartan distribution, too. That is
TaΦτ (K) ⊂ C(a) for any point a ∈ K.

The value of the 1-form κi on the vector field X(1) is κi(X
(1)) = ui

1,0 + Aui
0,1 for

i = 1, . . . ,m. Therefore, the restriction κi(X
(1)|E) = κi(X

(1))|E = 0 and the integral curves
of the vector field X(1)|E are integral for the Cartan distribution as well.

Let a ∈ L be an arbitrary point. The tangent space TaL is the direct sum

TaL = TaΦτ (K)⊕ RX(1)
a ⊂ C(a).

Therefore, L is a 2-dimensional integral manifold of the Cartan distribution.

4. EXAMPLE

Consider the system {
ut + uux = 0,
vt + uvx = 0

(4.5)

with initial data
u|t=0 =

1

1 + x2
, v|t=0 =

1

1 + (x− 1)2
.

Then the vector field
X = − ∂

∂t
− u0,0

∂

∂x

and its prolongation is

X(1) = − ∂

∂t
− u0,0

∂

∂x
+ u1,0u0,1

∂

∂u1,0

+ u2
0,1

∂

∂u0,1

+ u1,0v0,1
∂

∂v1,0
+ u0,1v0,1

∂

∂v0,1
.

The corresponding flow is

Φτ :



t 7→ t− τ,
x 7→ x− u0,0τ,

u0,0 7→ u0,0,

v0,0 7→ v0,0,

u1,0 7→
u1,0

1− τu0,1

u0,1 7→
u0,1

1− τu0,1

v1,0 7→
u1,0v0,1τ − v1,0u0,1τ + v1,0

1− τu0,1

,

v0,1 7→
v0,1

1− τu0,1

.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



54 E. KUSHNER

The Cauchy curve is

K :



t = 0,
x− χ = 0,

u0,0 −
1

1 + χ2
= 0,

v0,0 −
1

1 + (χ− 1)2
= 0,

u1,0 −
2χ

(1 + χ2)3
= 0,

v1,0 −
2χ− 2

(1 + χ2)(1 + (χ− 1)2)2
= 0,

u0,1 +
2χ

(1 + χ2)2
= 0,

v0,1 +
2χ− 2

(1 + (χ− 1)2)2
= 0.

Shifting the curve K along the vector field X(1), we obtain the multivalued solution to the
Cauchy problem:

LK :



t = τ,

x =
τ + χ+ χ3

1 + χ2
,

u0,0 =
1

1 + χ2
,

v0,0 =
1

2 + χ2 − 2χ
,

u1,0 =
2χ

(1 + χ2)(1 + 2χ2 + χ4 − 2τχ)
,

u0,1 = − 2χ

1 + 2χ2 + χ4 − 2τχ
,

v1,0 =
2χ3 − 2χ2 + 2χ− 2

(2 + χ2 − 2χ)2(1 + 2χ2 + χ4 − 2τχ)
,

v0,1 = − 2(χ− 1))(1 + χ2)2

(2 + χ2 − 2χ)2(1 + 2χ2 + χ4 − 2τχ)
.

The projections of this solution to 3-dimensional spaces t, x, u and t, x, v are shown in
Fig. 4.1.

Let us construct caustics. For this goal, we introduce two planes Π and Γ with coordinates
t, x and τ, χ respectively and construct the mapping π : Γ −→ Π, where

π : t = τ, x =
χ3 + χ+ τ

1 + χ2
.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



MULTIVALUED SOLUTIONS OF THE CAUCHY PROBLEM... 55

Fig. 4.1. Projections Lu
K and Lv

K of the multivalued solution LK to the spaces t, x, u and t, x, v respectively.

The Jacobi matrix of π is

Jπ =

 1 0
1

1 + χ2

1 + 2χ2 + χ4 − 2χτ

(1 + χ2)2

 .

A caustic Σ is defined by the critical points of this mapping, that is, the points at which the
determinant of the Jacobi matrix vanishes (see Fig. 4.2):

Σ = {1 + 2χ2 + χ4 − 2χτ = 0} ⊂ Γ.

Fig. 4.2. The caustics Σ on the plane Γ (left) and its projection to the plane Π (right).

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



56 E. KUSHNER

Fig. 4.3. Birth of the self-intersection curve: sections of the surface Lv
K at t = 0, t = 2, t = 5 respectively.

As can be seen from Fig. 4.1, the surface

Lv
K :


t = τ,

x =
τ + χ+ χ3

1 + χ2
,

v0,0 =
1

2 + χ2 − 2χ
,

(4.6)

has self-intersection along some curve. Let us find this curve. Coordinates a = a(τ) and
b = b(τ) of the points of this curve are solutions of the system

τ + a+ a3

1 + a2
=

τ + b+ b3

1 + b2
,

1

2 + a2 − 2a
=

1

2 + b2 − 2b
.

Its real solutions are

a = 1∓ (2τ − 4)1/4, b = 1± (2τ − 4)1/4

We see that self-intersection occurs only when τ ≥ 2. So, we can choose

χ = 1 + (2τ − 4)1/4.

Substituting this value to (4.6), we obtain a parametric representation of the self-intersection
curve: 

t = τ,

x =
τ + (2τ − 4)1/4 + 1 + ((2τ − 4)1/4 + 1)3

1 + ((2τ − 4)1/4 + 1)2
,

v0,0 =
1√

2τ − 4 + 1
.

5. CONCLUSION

The proposed method can be extended to a wider class of evolutionary systems. Note that
another approach to evolutionary systems is proposed in [3–5]. That method is based on the
finite-dimensional dynamics.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



MULTIVALUED SOLUTIONS OF THE CAUCHY PROBLEM... 57

Fig. 4.4. Projection of the self-intersection curve to the plane x, v

REFERENCES

1. Krasilshchik, I. S., Lychagin, V. V. & Vinogradov, A. M. (1986) Geometry of jet spaces
and nonlinear partial differential equations. New York, NY: Gordon and Breach.

2. Kushner, A. G., Lychagin, V. V. & Rubtsov, V. N. (2007) Contact geometry and
nonlinear differential equations. Cambridge, UK: Cambridge University Press.

3. Gorinov, A. A. & Kushner, A. G. (2020) Dynamics of evolutionary PDE systems,
Lobachevskii J. Math., 41, 2448–2457.

4. Kushner, A. G. & Matviichuk, R. I. (2021) Dynamics and exact solutions of non-
evolutionary partial differential equations, Differ. Geom. Appl., 76, 101761.

5. Kushner, A. G. & Sinian, T. (2024) Evolutionary Systems and Flows on Solutions
Spaces of Finite Type Equations, Lobachevskii J. Math., 44(9), 3945–3951.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)


	Introduction
	Multivalued Solutions
	Cauchy Data
	Example
	Conclusion