A system of ODEs for a Perturbation of a Minimal Mass Soliton

TL;DR

This paper derives a system of ODEs to model the short-term dynamics of perturbed minimal mass solitons in a nonlinear Schrödinger equation with saturated nonlinearity, supported by numerical evidence of two possible outcomes.

Contribution

It introduces a novel ODE system capturing the behavior of small perturbations around minimal mass solitons in a saturated nonlinear Schrödinger equation.

Findings

Perturbed solitons oscillate around stable families.

Unstable initial data disperses following the unstable soliton branch.

Numerical simulations support the two dynamical outcomes.

Abstract

We study soliton solutions to a nonlinear Schrodinger equation with a saturated nonlinearity. Such nonlinearities are known to possess minimal mass soliton solutions. We consider a small perturbation of a minimal mass soliton, and identify a system of ODEs similar to those from Comech and Pelinovsky (2003), which model the behavior of the perturbation for short times. We then provide numerical evidence that under this system of ODEs there are two possible dynamical outcomes, which is in accord with the conclusions of Pelinovsky, Afanasjev, and Kivshar (1996). For initial data which supports a soliton structure, a generic initial perturbation oscillates around the stable family of solitons. For initial data which is expected to disperse, the finite dimensional dynamics follow the unstable portion of the soliton curve.