Nonexistence of soliton-like solutions for defocusing generalized KdV equations

TL;DR

This paper proves that defocusing generalized KdV equations do not admit soliton-like solutions under decay assumptions, using Tao's theorem to establish a dispersion estimate.

Contribution

It introduces a novel dispersion estimate based on Tao's theorem to demonstrate the nonexistence of soliton-like solutions for these equations.

Findings

No soliton-like solutions with decay exist for defocusing generalized KdV.

Dispersion estimate derived from Tao's theorem is key to the proof.

The result advances understanding of the global dynamics of these equations.

Abstract

In this short note, we consider the global dynamics of the defocusing generalized KdV equations: u_t + u_{xxx} = (|u|^{p-1}u)_x. We use Tao's theorem that the energy moves faster than mass to prove a moment type dispersion estimate. As an application of the dispersion estimate, we show that there is no soliton-like solutions with decaying assumption.