Dispersive limit from the Kawahara to the KdV equation

Abstract

We investigate the limit behavior of the solutions to the Kawahara equation $$u_t+u_{3x}+\epsilon u_{5x}+u{u}_x=0,$$ as $0<\epsilon \to 0$. In this equation, the terms $u_{3x}$ and $\epsilon u_{5x}$ do compete together and do cancel each other at frequencies of order $1/\sqrt{\epsilon }$. 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 $C([0,T];H^1(\mathbb{R}))$ towards the solutions of the KdV equation for any fixed $T>0$.

arXiv commentThere was something incorrect in the section 3 of the first version. This version is corrected