Liapunov–Schmidt Reduction and Continuation for Nonlinear Schrödinger Equations

Abstract

We study the bifurcation scenario of nonlinear Schrödinger equations (NLS). The Liapunov–Schmidt reduction is applied to show that the simple bifurcations of a single NLS are pitchfork. The pitchfork bifurcation can be subcritical or supercritical, depending on the coefficient of the cubic term we choose. We also describe numerical methods so that the Liapunov–Schmidt reduction can effectively handle a corank‐2 bifurcation point. Next, we apply numerical continuation methods to trace solution curves and surfaces of the NLS, where the system is discretized by the centered difference approximations. Numerical results on two‐ and three‐dimensional M‐coupled NLS are reported, where the physical properties such as the effect of trapping potentials, isotropic and nonisotropic trapping potentials, mass conservation constraints, and strong and weak repulsive interactions are considered in our numerical experiments.