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Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity

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Abstract

We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schrödinger equations with nonzero conditions at infinity (includindg the Gross–Pitaevskii and the so-called “cubic-quintic” equations) in space dimension \({ N \geq 2}\). We show that minimization of the energy at fixed momentum can be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves, while minimization of the action at constant kinetic energy can be used in all cases. We also explore the relationship between the families of traveling waves obtained by different methods and we prove a sharp nonexistence result for traveling waves with small energy.

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Authors and Affiliations

  1. Laboratoire J.A. Dieudonné UMR 6621, Université de Nice-Sophia Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    David Chiron

  2. Institut de Mathématiques de Toulouse UMR 5219, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Mihai Mariş

  3. Institut Universitaire de France, Paris, France

    Mihai Mariş

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  1. David Chiron
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Correspondence to Mihai Mariş.

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Communicated by P. Rabinowitz

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Chiron, D., Mariş, M. Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity. Arch Rational Mech Anal 226, 143–242 (2017). https://doi.org/10.1007/s00205-017-1131-2

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