Collapse of solitary waves near transition from supercritical to subcritical bifurcations

TL;DR

This paper investigates the nonlinear evolution and collapse of one-dimensional solitons near the transition from supercritical to subcritical bifurcations, revealing self-similar behavior and asymmetries in pulse dynamics across various physical systems.

Contribution

It provides a combined analytical and numerical analysis of soliton instability near bifurcation transition points, highlighting self-similar collapse behavior and tail asymmetries.

Findings

Pulse amplitude and width exhibit self-similar collapse near the transition.

Asymmetry in pulse tails arises due to self-steepening effects.

Results apply to water waves and optical pulses, demonstrating broad relevance.

Abstract

We study both analytically and numerically the nonlinear stage of the instability of one-dimensional solitons in a small vicinity of the transition point from supercritical to subcritical bifurcations in the framework of the generalized nonlinear Schr\"{o}dinger equation. It is shown that near the collapsing time the pulse amplitude and its width demonstrate the self-similar behavior with a small asymmetry at the pulse tails due to self-steepening. This theory is applied to both solitary interfacial deep-water waves and envelope water waves with a finite depth and short optical pulses in fibers as well.