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Solitary Waves for Nonlocal Derivative Nonlinear Schrödinger Equations: A Variational Characterization

Amin Esfahani, Adilbek Kairzhan, Mukhtar Karazym

Abstract

We establish several existence results for traveling-wave solutions of the nonlocal derivative nonlinear Schrödinger equation by means of the Nehari manifold technique and the concentration-compactness lemma. We study associated minimization problems in the subcritical and critical cases and prove the existence of at least one minimizer in each case. In the critical case, we additionally obtain explicit solitary-wave solutions for a family of parameters. Finally, we derive Pohozaev-type identities and use them to establish corresponding nonexistence results.

Solitary Waves for Nonlocal Derivative Nonlinear Schrödinger Equations: A Variational Characterization

Abstract

We establish several existence results for traveling-wave solutions of the nonlocal derivative nonlinear Schrödinger equation by means of the Nehari manifold technique and the concentration-compactness lemma. We study associated minimization problems in the subcritical and critical cases and prove the existence of at least one minimizer in each case. In the critical case, we additionally obtain explicit solitary-wave solutions for a family of parameters. Finally, we derive Pohozaev-type identities and use them to establish corresponding nonexistence results.
Paper Structure (14 sections, 28 theorems, 349 equations, 1 table)

This paper contains 14 sections, 28 theorems, 349 equations, 1 table.

Table of Contents

  1. Introduction
  2. The model
  3. Main result and motivation
  4. Related results
  5. Structure of the paper
  6. Notation and preliminaries
  7. Gauge transformation and traveling wave ansatz
  8. Subcritical minimization on the Nehari manifold
  9. Critical minimization on the Nehari manifold
  10. Subcritical Minimization with Fixed Nonlocal Mean Flow and a Quartic Nonlinearity
  11. Critical Minimization with a Fixed Quartic Nonlinearity
  12. Nonexistence of traveling wave solutions
  13. Acknowledgments
  14. Data availability

Key Result

Lemma 1.1

Let $\{\rho_n\}$ be a sequence of nonnegative functions in $L^1(\mathbb{R},\mathbb{R})$ such that Then, up to a subsequence, exactly one of the following alternatives occurs:

Theorems & Definitions (52)

  • Lemma 1.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 42 more