Physica D 241 (2012) 125–133
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Physica D
journal homepage: www.elsevier.com/locate/physd
The Symmetric Regularized-Long-Wave equation: Well-posedness and
nonlinear stability
Carlos Banquet Brango
∗
Departamento de Matemáticas y Estadística, Universidad de Córdoba, Carrera 6 No. 76-103, Montería, Córdoba, Colombia
article info
Article history:
Received 4 May 2011
Received in revised form
17 October 2011
Accepted 19 October 2011
Available online 26 October 2011
Communicated by J. Bronski
Keywords:
Periodic travelling waves
Nonlinear stability
Symmetric Regularized Long Wave
equation
Well-posedness
abstract
The focus of the present work is the Symmetric Regularized-Long-Wave equation. We prove that the initial
value problem for this equation is locally and globally well-posed in H
s
per
× H
s−1
per
and H
s
(R) × H
s−1
(R), if
s ≥ 0. We also prove the existence and nonlinear stability of periodic travelling wave solutions, of cnoidal
type, for the equation mentioned above.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
In this paper, we study different aspects of the Symmetric
Regularized-Long-Wave equation (SRLW equation henceforth)
u
tt
− u
xx
+ (uu
x
)
t
− u
xxtt
= 0, (1.1)
where u is a real-valued function and the subscripts denote the
derivative with respect to the spatial variable x and time t . The
Symmetric Regularized-Long-Wave equation is a model for the
weakly nonlinear ion acoustic and space-charge waves. This equa-
tion was introduced by Seyler and Fenstermacher in [1], where
a weakly nonlinear analysis of the cold-electron fluid equation is
made.
Eq. (1.1) has the equivalent form
u
t
− u
xxt
+ uu
x
− v
x
= 0,
v
t
− u
x
= 0,
(1.2)
for all t > 0 and x ∈ R. This equivalent system has four conserva-
tion laws
∗
Tel.: +57 3145933379.
E-mail addresses: cabanquet@hotmail.com,
cbanquet@correo.unicordoba.edu.co.
E (u,v) =
∫
uv −
1
6
u
3
dx,
V (u,v) =
1
2
∫
(u
2
+ u
2
x
+ v
2
)dx,
I
1
(u,v) =
∫
u dx and I
2
(u,v) =
∫
v dx.
(1.3)
The first aspect that we study about the SRLW equation is
the local and global well-posedness. On the periodic case, we
prove that Eq. (1.2) with initial data (u
0
,v
0
) ∈ H
s
per
([−L, L]) ×
H
s−1
per
([−L, L]) is globally well-posed if s ≥ 0. The more important
ingredient here is a bilinear estimative obtained by Roumégoux
in [2]. On the continuous case (this is for initial data in H
s
(R) ×
H
s−1
(R)), we used the ideas given in [3] to prove that (1.2) with
initial data (u
0
,v
0
) ∈ H
s
(R) × H
s−1
(R) is globally well-posed if
s ≥ 0. This result improves the theory established by Chen [4],
where a global well-posedness result was proved in H
1
(R) ×L
2
(R).
Another important part of this paper is to establish the
existence and nonlinear stability of periodic travelling wave
solutions for Eq. (1.1). It is well known that the study of this kind
of states of motion is very important to understand the behaviour
of many physical systems. In the past two decades, various results
about nonlinear stability for some equations and systems have
been obtained; see for example [5–24].
In this work, we are interested in giving a stability theory of
periodic travelling wave solutions for the nonlinear system (1.2).
0167-2789/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.physd.2011.10.007