Dispersive limit from the Kawahara to the KdV equation

TL;DR

This paper studies the limit of solutions to the Kawahara equation as the dispersive parameter approaches zero, showing convergence to the KdV equation despite challenges from competing dispersive terms.

Contribution

It introduces a novel combination of dispersive approaches to prove convergence in the dispersive limit of the Kawahara to the KdV equation.

Findings

Solutions converge in C([0,T];H^1(R)) as epsilon approaches zero.

Standard dispersive methods are insufficient due to competing terms.

The approach successfully handles frequency interactions at order 1/√epsilon.

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$.