A new family of solitons for nonlinear Schr\"odinger equations with non-vanishing boundary conditions in high dimension

Abstract

In space dimensions $N \geq 4$, we introduce a new minimization procedure to construct traveling wave solutions to nonlinear Schr\"odinger equations with non-vanishing boundary conditions at spatial infinity. We denote the family of solitons obtained using this construction by $\mathscr{J}$. Mari\c{s} (Ann. of Math. 178:107-182, 2013) obtained a family of solitons by minimizing the action functional subject to a Pohozaev constraint; we use $\mathscr{P}$ to denote this family of solitons. Chiron and Mari\c{s} (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 $\mathscr{Q}$. We show that, under some conditions, we have $\mathscr{Q} \subset \mathscr{J} \subset \mathscr{P}$. In addition, we show that $\mathscr{P} \subset \mathscr{J}$ under specific conditions.