Numerical Computations for Bifurcations and Spectral Stability of Solitary Waves in Coupled Nonlinear Schr\"odinger Equations

TL;DR

This paper numerically investigates the bifurcation structure and spectral stability of solitary waves in coupled nonlinear Schrödinger equations, confirming recent theoretical predictions and analyzing stability changes at bifurcation points.

Contribution

It provides the first detailed numerical analysis of bifurcations and spectral stability in coupled nonlinear Schrödinger equations, including eigenfunction computations.

Findings

Detection of pitchfork bifurcations of solitary waves

Eigenvalue analysis confirming stability predictions

Stability of bifurcated waves remains unchanged at saddle-node bifurcation

Abstract

We numerically study solitary waves in the coupled nonlinear Schr\"odinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change its stability type at the saddle-node bifurcation point.