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2005 Stability of dnoidal waves to Hirota-Satsuma system
Jaime Angulo Pava
Differential Integral Equations 18(6): 611-645 (2005).


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.


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Jaime Angulo Pava. "Stability of dnoidal waves to Hirota-Satsuma system." Differential Integral Equations 18 (6) 611 - 645, 2005.


Published: 2005
First available in Project Euclid: 21 December 2012

zbMATH: 1201.76033
MathSciNet: MR2136702

Primary: 35Q53
Secondary: 35B10 , 35B35 , 35Q51 , 37K40 , 37K45 , 76B25

Rights: Copyright © 2005 Khayyam Publishing, Inc.


Vol.18 • No. 6 • 2005
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