Orbital stability of black solitons for quasilinear Schr\"odinger equations with nonzero conditions at infinity

TL;DR

This paper proves the orbital stability of black solitons in a class of quasilinear Schrödinger equations with nonzero boundary conditions, using variational methods and the VK slope condition.

Contribution

It establishes the stability criteria for black solitons in quasilinear Schrödinger equations, including explicit verification of the VK condition and analysis of variational problems.

Findings

Black solitons are orbitally stable under the VK slope condition.

Explicit formula provided for verifying the VK condition.

Stability proven for a broad class of nonlinearities.

Abstract

We investigate the orbital stability of black solitons for a broad class of quasilinear Schr\"odinger equations in one space dimension, with nonzero boundary conditions at infinity. Namely, our framework handles general defocusing semilinear nonlinearities and focusing or defocusing quasilinear nonlinearities. First, we establish sufficient conditions on the quasi-linear nonlinearities ensuring the existence of a local branch of finite-energy solitons parameterized by their speed. Within this branch, the black soliton, also called kink, corresponds to the stationary solution. Our main result is the orbital stability of the black soliton in the energy space, provided that the Vakhitov-Kolokolov (VK) slope condition holds; namely, that the derivative of the momentum with respect to the speed is negative at zero. Moreover, we derive an explicit formula for verifying this VK condition. The proof relies on the analysis of a carefully designed variational problem, which allows us to control the sup-norm of the evolution of a perturbation of the kink in terms of the energy and momentum, both of which are conserved by the flow. A delicate part of the argument is the analysis of minimizing sequences for this variational problem, since the infimum is not attained.