Stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear Schrödinger equations

Abstract

Linear stability of both sign-definite (positive) and sign-indefinite solitary waves near pitchfork bifurcations is analyzed for the generalized nonlinear Schrödinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions. Bifurcations of linear-stability eigenvalues associated with pitchfork bifurcations are analytically calculated. Based on these eigenvalue-bifurcation formulae, linear stability of solitary waves near pitchfork bifurcations is then determined. It is shown that the base solution branch switches stability at the bifurcation point. In addition, the two bifurcated solution branches and the base branch have the opposite (same) stability when their power slopes have the same (opposite) sign. Furthermore, the stability of these solution branches can be determined almost exclusively from their power diagram (especially for positive solitary waves). These stability results are also compared with the Hamiltonian–Krein index theory, and they are shown to be consistent with each other. Lastly, various numerical examples are presented, and the numerical results confirm the analytical predictions both qualitatively and quantitatively.