No Stability Switching at Saddle-Node Bifurcations of Solitary Waves in Generalized Nonlinear Schroedinger Equations

TL;DR

This paper demonstrates that, contrary to previous beliefs, solitary waves in certain generalized nonlinear Schrödinger equations do not necessarily change stability at saddle-node bifurcations, supported by analytical proofs and numerical examples.

Contribution

It proves analytically that stability does not switch at saddle-node bifurcations for a broad class of generalized nonlinear Schrödinger equations, challenging prior assumptions.

Findings

Solitary waves can remain stable at saddle-node bifurcations.

Analytical proof applies to equations with real or complex potentials.

Numerical examples confirm stability persistence at bifurcations.

Abstract

Saddle-node bifurcations arise frequently in solitary waves of diverse physical systems. Previously it was believed that solitary waves always undergo stability switching at saddle-node bifurcations, just as in finite-dimensional dynamical systems. Here we show that this is not true. For a large class of generalized nonlinear Schr\"odinger equations with real or complex potentials, we prove that stability of solitary waves does not switch at saddle-node bifurcations. This analytical result is confirmed by numerical examples where both soliton branches are stable at saddle-node bifurcations.