A new family of solitons for nonlinear Schrödinger equations with non-vanishing boundary conditions in high dimension

Abstract

In space dimensions $N\ge 4$, we introduce a new minimization procedure to construct traveling wave solutions to nonlinear Schrödinger equations with non-vanishing boundary conditions at spatial infinity. We denote the family of solitons obtained using this construction by $\mathcal{J}$. Mariş (Ann. of Math. 178:107-182, 2013) obtained a family of solitons by minimizing the action functional subject to a Pohozaev constraint; we use $\mathcal{P}$ to denote this family of solitons. Chiron and Mariş (Arch. Rational Mech. Anal. 226:143-242, 2017) used minimizing energy at fixed momentum to obtain a family of solitons; we denote this family of solitons by $\mathcal{Q}$. We show that, under some conditions, we have $\mathcal{Q}\subset \mathcal{J}\subset \mathcal{P}$. In addition, we show that $\mathcal{P}\subset \mathcal{J}$ under specific conditions.