## Differential and Integral Equations

- Differential Integral Equations
- Volume 18, Number 6 (2005), 611-645.

### Stability of dnoidal waves to Hirota-Satsuma system

#### Abstract

This work is concerned with nonlinear stability properties of periodic travelling wave
solutions of the Hirota-Satsuma system $$ \begin{cases} u_t+ u_{xxx}+6u_xu=2bvv_x\\ v_t+
v_{xxx}+ 3uv_x=0 \end{cases} $$ posed in $\mathbb R$ with $b>0$. We prove that this system
is globally well posed in $L^2_{per}([0,L])\times H^1_{per}([0,L])$ by using Bourgain's
space framework. Also shown is the existence of at least two nontrivial smooth curves of
periodic travelling wave solutions depending on the classical Jacobian elliptic functions.
We find *dnoidal* and *cnoidal* waves solutions. Then we prove the nonlinear
stability of the dnoidal waves solutions in the energy space $L^2_{per}([0,L])\times
H^1_{per}([0,L])$. The Floquet theory is used to obtain a detailed spectral analysis of
the Jacobian form of Lam\'e's equation which is required in our stability theory.

#### Article information

**Source**

Differential Integral Equations, Volume 18, Number 6 (2005), 611-645.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060173

**Mathematical Reviews number (MathSciNet)**

MR2136702

**Zentralblatt MATH identifier**

1201.76033

**Subjects**

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Secondary: 35B10: Periodic solutions 35B35: Stability 35Q51: Soliton-like equations [See also 37K40] 37K40: Soliton theory, asymptotic behavior of solutions 37K45: Stability problems 76B25: Solitary waves [See also 35C11]

#### Citation

Angulo Pava, Jaime. Stability of dnoidal waves to Hirota-Satsuma system. Differential Integral Equations 18 (2005), no. 6, 611--645. https://projecteuclid.org/euclid.die/1356060173