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hal-00694082, version 1

Dispersive limit from the Kawahara to the KdV equation

Luc Molinet () 1, Yuzhao Wang () 2

Abstract: We investigate the limit behavior of the solutions to the Kawahara equation $$u_t +u_{3x} +\varepsilon u_{5x} + u u_x =0 ,$$ as $0<\varepsilon \to 0$. In this equation, the terms $u_{3x}$ and $\varepsilon u_{5x}$ do compete together and do cancel each other at frequencies of order $1/\sqrt{\varepsilon}$. This prohibits the use of a standard dispersive approach for this problem. Nervertheless, by combining different dispersive approaches according to the range of spaces frequencies, we succeed in proving that the solutions to this equation converges in $C([0,T];H^1(\R))$ towards the solutions of the KdV equation for any fixed $T>0$.

• 1:  Laboratoire de Mathématiques et Physique Théorique (LMPT)
• CNRS : UMR6083 – Université François Rabelais - Tours
• 2:  Department of Mathematics and Physics
• North China Electric Power University
• Domain : Mathematics/Analysis of PDEs
• Keywords : KdV equation – Kawahara equation – dispersive limit
• Available versions :  v1 (2012-05-03) v2 (2012-06-07)

• hal-00694082, version 1
• oai:hal.archives-ouvertes.fr:hal-00694082
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• Submitted on: Thursday, 3 May 2012 15:08:03
• Updated on: Thursday, 3 May 2012 17:12:26