Persistence of kinks in the NLS equation with competing nonlinearities

Abstract

External potentials break the translational invariance of nonlinear Schrödinger (NLS) equations and generally convert a continuous family of standing wave solutions into isolated pinned states. We study this mechanism for the prototypical case of a kink in the NLS equation with competing focusing cubic and defocusing quintic nonlinearities. The Lyapunov--Schmidt reduction method is used to prove that the location $s_0$ of the pinned kinks is selected near roots of an effective potential $V_{\rm eff}(s)$. We then determine how the translational zero eigenvalue of the Jacobian operator splits under the perturbation and prove that the sign of $V_{\rm eff}'(s_0)$ distinguishes eigendirections of positive and negative energy. We further show that if the Jacobian operator admits a negative eigenvalue, then the stability problem contains a real unstable eigenvalue. The derivation of the asymptotic expansion for the unstable eigenvalue is nonstandard because the zero eigenvalue is embedded in the continuous spectrum of the linearized operator. Finally, we show that the persistence of kinks lacks parity symmetry even if the external potential is odd. Numerical continuation, spectral computations with domains adapted to the slowly decaying eigenfunctions, and direct dynamical simulations of the time evolution corroborate the analytical predictions for different representative examples.