Instability of solitons - revisited, I: the critical generalized KdV equation

TL;DR

This paper revisits the instability of solitons in the critical generalized KdV equation, offering simplified methods to establish instability that could be applicable to other KdV-type equations.

Contribution

It introduces simplified approaches using truncation and monotonicity to prove soliton instability in the critical case, improving upon previous complex decay estimate methods.

Findings

Confirmed soliton instability for p ≥ 5 in the critical case

Developed simplified proof techniques applicable to other KdV equations

Enhanced understanding of soliton dynamics at critical nonlinearity

Abstract

We revisit the phenomenon of instability of solitons in the generalized Korteweg-de Vries equation, $u_t + \partial_x(u_{xx} + u^p) = 0$. It is known that solitons are unstable for nonlinearities $p \geq 5$, with the critical power $p=5$ being the most challenging case to handle. The critical case was proved by Martel-Merle in [11], where the authors crucially relied on the pointwise decay estimates of the linear KdV flow. In this paper, we show simplified approaches to obtain the instability of solitons via truncation and monotonicity, which can be also useful for other KdV-type equations.