The phase shift of line solitons for the KP-II equation

TL;DR

This paper investigates the local phase shift of line solitons in the KP-II equation, providing bounds for the phase shift's behavior under localized perturbations, which enhances understanding of soliton stability.

Contribution

It establishes an $L^$-bound for the local phase shift of modulating line solitons in the KP-II equation with localized perturbations, a novel stability analysis result.

Findings

Bounded the local phase shift in $L^$ norm.

Demonstrated non-uniformity of phase shift in transverse direction.

Enhanced understanding of soliton stability under perturbations.

Abstract

The KP-II equation was derived by [B. B. Kadomtsev and V. I. Petviashvili,Sov. Phys. Dokl. vol.15 (1970), 539-541] to explain stability of line solitary waves of shallow water. Stability of line solitons has been proved by [T. Mizumachi, Mem. of vol. 238 (2015), no.1125] and [T. Mizumachi, Proc. Roy. Soc. Edinburgh Sect. A. vol.148 (2018), 149--198]. It turns out the local phase shift of modulating line solitons are not uniform in the transverse direction. In this paper, we obtain the $L^\infty$-bound for the local phase shift of modulating line solitons for polynomially localized perturbations.