Differential and Integral Equations

Stability of dnoidal waves to Hirota-Satsuma system

Jaime Angulo Pava

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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

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

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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

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