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

TL;DR

This paper constructs and characterizes solutions to supercritical gKdV equations that converge to solitons over time, revealing new types of solutions beyond exact solitons using spectral analysis.

Contribution

It introduces a method to construct and classify solutions converging to solitons in the supercritical case, where such solutions are not necessarily solitons.

Findings

Existence of solutions converging to solitons without being solitons

Construction of a one-parameter family of special solutions

Spectral analysis of the linearized operator around solitons

Abstract

We consider the generalized Korteweg-de Vries equation in the supercritical case, and we are interested in solutions which converge to a soliton in large time in H^1. In the subcritical case, such solutions are forced to be exactly solitons by variational characterization, 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 due to Pego and Weinstein, we construct a one parameter family of special solutions which characterizes all such special solutions.