Conditions and Stability Analysis for Saddle-Node Bifurcations of Solitary Waves in Generalized Nonlinear Schrödinger Equations

Saddle-node bifurcations of solitary waves in generalized nonlinear Schrödinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions are analyzed. First, general conditions for these bifurcations are derived. Second, it is shown analytically that the linear stability of these solitary waves does not switch at saddle-node bifurcations, which is in stark contrast with finite-dimensional dynamical systems where stability switching takes place. Third, it is shown that this absence of stability switching does not contradict the Vakhitov–Kolokolov stability criterion or the results in finite-dimensional dynamical systems. Fourth, it is shown that this absence of stability switching holds not only for real potentials but also for complex potentials. Lastly, various numerical examples will be given to confirm these analytical findings. In particular, saddle-node bifurcations with both branches of solitary waves being stable will be presented.