Unique and nonunique minimizing travelling waves for some Ginzburg–Landau type equations

This is a summary of an upcoming work in collaboration with Mihai Maris. We study the Schrödinger equation with a nonzero condition at infinity for some nonlinearities, for which the existence of travelling waves has been proven in [22] as solutions of a minimization problem. We show that for some of them, we do not have uniqueness of this minimizer, and we study how these travelling changes when we change the speed and/or the nonlinearity. We also discuss some examples where we have the uniqueness of travelling waves that are solutions to minimisation problems, as well as questions related to the stability of these objects.