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Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity. (English) Zbl 1315.35207
Let $$c$$ denote any speed less than the sound velocity, and $$F$$ is a function $$\mathbb{R}_+ \mapsto \mathbb{R}$$. The purpose of this article is to study the NLS equation $-ic\frac{\partial u}{\partial x_1}+\triangle u +F(| 1+u | ^2)(1+u) =0,\;u\in \mathbb{R}^N. \tag{v}$ Formally, the solutions of ({v}) are the critical points of a certain functional. Separate proofs are given for $$N\geq 4$$ and $$N=3$$. The proofs use Sobolev embedding, Hölder’s inequality, Fatou’s lemma, Fubini’s theorem, Sobolev’s and Gagliardo’s inequalities, Morrey’s inequality. The Poincaré inequality from the theory of Sobolev spaces, the dominated convergence theorem, and Lagrange multipliers.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35C07 Traveling wave solutions
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