Dynamics near a minimal-mass soliton for a Korteweg-de Vries equation

TL;DR

This paper investigates the dynamics near a minimal-mass soliton in a generalized Korteweg-de Vries equation with saturated nonlinearity, deriving a simple ODE model that reveals non-oscillatory behavior and potential instability.

Contribution

It introduces a new ODE system modeling perturbations near a minimal-mass soliton in a KdV equation with saturated nonlinearity, highlighting unique non-oscillatory dynamics.

Findings

Identified a simple dynamical system with a single unstable fixed point

Demonstrated non-oscillatory behavior distinguishes KdV from NLS

Connected the ODE model to previous theoretical work

Abstract

We study soliton solutions to a generalized Korteweg - de Vries (KdV) equation with a saturated nonlinearity, following the line of inquiry of the authors for the nonlinear Schr\"odinger equation (NLS). KdV with such a nonlinearity is known to possess a minimal-mass soliton. We consider a small perturbation of a minimal-mass soliton and identify a system of ODEs, which models the behavior of the perturbation for short times. This connects nicely to a work of Comech, Cuccagna & Pelinovsky (2007). These ODEs form a simple dynamical system with a single unstable hyperbolic fixed point with two possible dynamical outcomes. A particular feature of the dynamics are that they are non-oscillatory. This distinguishes the KdV problem from the analogous NLS one.