Classification of solitary wave bifurcations in generalized nonlinear Schr\"odinger equations

TL;DR

This paper classifies and analyzes bifurcations of solitary waves in generalized nonlinear Schrödinger equations, providing analytical conditions, shape descriptions of power diagrams, and numerical examples including the first report of transcritical bifurcations.

Contribution

It introduces a comprehensive classification of solitary wave bifurcations in generalized nonlinear Schrödinger equations with analytical conditions and novel numerical demonstrations.

Findings

Analytical conditions for saddle-node, pitchfork, and transcritical bifurcations.

Distinct power diagram shapes near different bifurcations.

First numerical report of transcritical bifurcations in this context.

Abstract

Bifurcations of solitary waves are classified for the generalized nonlinear Schr\"odinger equations with arbitrary nonlinearities and external potentials in arbitrary spatial dimensions. Analytical conditions are derived for three major types of solitary wave bifurcations, namely saddle-node bifurcations, pitchfork bifurcations and transcritical bifurcations. Shapes of power diagrams near these bifurcations are also obtained. It is shown that for pitchfork and transcritical bifurcations, their power diagrams look differently from their familiar solution-bifurcation diagrams. Numerical examples for these three types of bifurcations are given as well. Of these numerical examples, one shows a transcritical bifurcation, which is the first report of transcritical bifurcations in the generalized nonlinear Schr\"odinger equations. Another shows a power loop phenomenon which contains several saddle-node bifurcations, and a third example shows double pitchfork bifurcations. These numerical examples are in good agreement with the analytical results.