Dispersive limit from the Kawahara to the KdV equation

We investigate the limit behavior of the solutions to the Kawahara equationut+u3x+εu5x+uux=0,ε>0 as ε→0ε→0. In this equation, the terms u3xu3x and εu5xεu5x compete each other and cancel each other at frequencies of order 1/ε. This prohibits the use of a standard dispersive approach for this problem. Nevertheless, by combining different dispersive approaches according to the space frequency ranges, we succeed in proving that the solutions to this equation converge in C([0,T];H1(R))C([0,T];H1(R)) towards the solutions of the KdV equation for any fixed T>0T>0.