Construction and characterization of solutions converging to solitons for supercritical gKdV equations

Abstract

We consider the generalized Korteweg-de Vries equation $$ \partial_t u +\partial_x^3 u +\partial_x(u^p)=0, \quad (t,x)\in{\mathbb{R}}^2, $$ in the supercritical case $p>5$, and we are interested in solutions which converge to a soliton in large time in $H^1$. In the subcritical case ($p < 5$), such solutions are forced to be exactly solitons by variational characterization [3, 32], but no such result exists in the supercritical case. In this paper, we first construct a ``special solution" in this case by a compactness argument, i.e., a solution which converges to a soliton without being a soliton. Secondly, using a description of the spectrum of the linearized operator around a soliton [20], we construct a one-parameter family of special solutions which characterizes all such special solutions. In the case of the nonlinear Schrödinger equation, a similar result was proved in [7, 8].