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

Xiuqing Duan

Abstract

In space dimensions $N \geq 4$, 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 $\mathscr{J}$. 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 $\mathscr{P}$ 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 $\mathscr{Q}$. 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.

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

Abstract

In space dimensions , 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 . 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 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 . We show that, under some conditions, we have . In addition, we show that under specific conditions.
Paper Structure (6 sections, 24 theorems, 293 equations, 1 figure)

This paper contains 6 sections, 24 theorems, 293 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Approximation Process
  3. Existence in dimension $N \geq 4$
  4. Dimension N=3
  5. Symmetry of the solutions
  6. Relationship between different families of solitons

Key Result

Theorem 1.1

Suppose $N \geq 4$, $0<\nu<\sqrt{2}$, $(Ass1)$, and $(Ass2)$ are satisfied. Then: (i) For any $j>0$, $\{w \in \mathcal{E} ~|~ J(w)=j>0\}$ is not empty. (ii) Let $\{w_n\}_{n \geq 1} \subset \mathcal{E}$ satisfy Then up to subsequence and translation, there exists $w \in \mathcal{E}$ such that and we have That is, $w$ minimizes $T$ under $J(w)=j$. (iii) Let $w$ minimize $T$ in the set $\{w \in \m

Figures (1)

  • Figure 1: E-P relation for the GP equation. $N=2$ in the left figure, $N=3$ in the right figure.

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 21 more