Stability switching at transcritical bifurcations of solitary waves in generalized nonlinear Schroedinger equations

TL;DR

This paper analyzes the linear stability of solitary waves near transcritical bifurcations in generalized nonlinear Schrödinger equations, revealing stability switching and eigenvalue bifurcations, with results confirmed numerically.

Contribution

It provides an analytical calculation of eigenvalue bifurcations and stability switching in generalized nonlinear Schrödinger equations at transcritical bifurcations, extending understanding beyond finite-dimensional systems.

Findings

Both solution branches undergo stability switching at the bifurcation point.

The two branches have opposite linear stability.

Analytical results agree well with numerical simulations.

Abstract

Linear stability of solitary waves near transcritical bifurcations is analyzed for the generalized nonlinear Schroedinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions. Bifurcation of linear-stability eigenvalues associated with this transcritical bifurcation is analytically calculated. Based on this eigenvalue bifurcation, it is shown that both solution branches undergo stability switching at the transcritical bifurcation point. In addition, the two solution branches have opposite linear stability. These stability properties closely resemble those for transcritical bifurcations in finite-dimensional dynamical systems. This resemblance for transcritical bifurcations contrasts those for saddle-node and pitchfork bifurcations, where stability properties in the generalized nonlinear Schroedinger equations differ significantly from those in finite-dimensional dynamical systems. The analytical results are also compared with numerical results, and good agreement is obtained.