On stabilization at a soliton for generalized Korteweg--De Vries pure power equation for any power $p\in (1,5)$

TL;DR

This paper proves asymptotic stability of solitons for the generalized KdV equation with power $p$ in (1,5), using a novel combination of virial and energy inequalities, extending previous methods from NLS analysis.

Contribution

It introduces a new approach combining virial and Kato smoothing inequalities to establish stability for generalized KdV solitons across a range of powers.

Findings

Proves asymptotic stability of solitons for p in (1,5).

Develops a new method combining virial and energy inequalities.

Extends stability results from NLS to generalized KdV equations.

Abstract

We apply our idea, which previously we used in the analysis of the pure power NLS, consisting in spitting the virial inequality method into a large energy inequality combined with Kato smoothing, to the case of generalized Korteweg--De Vries pure power equations. We assume that a solution remains for all positive times very close to a soliton and then we prove an asymptotic stability result for $t\to +\infty$.