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Variational properties of space-periodic standing waves of nonlinear Schr{ö}dinger equations with general nonlinearities

Perla Kfoury, Stefan Le Coz

Abstract

Periodic waves are standing wave solutions of nonlinear Schr\''odinger equations whose profile is periodic in space dimension one. We consider general nonlinearities and provide variational characterizations for the periodic wave profiles. This involves minimizing energy while keeping mass and momentum constant, as well as minimizing the action over the Nehari manifold. These variational approaches are considered both in the periodic and anti-periodic settings, and for focusing and defocusing nonlinearities. In appendix, we study the existence properties of periodic solutions of the triple power nonlinearity.

Variational properties of space-periodic standing waves of nonlinear Schr{ö}dinger equations with general nonlinearities

Abstract

Periodic waves are standing wave solutions of nonlinear Schr\''odinger equations whose profile is periodic in space dimension one. We consider general nonlinearities and provide variational characterizations for the periodic wave profiles. This involves minimizing energy while keeping mass and momentum constant, as well as minimizing the action over the Nehari manifold. These variational approaches are considered both in the periodic and anti-periodic settings, and for focusing and defocusing nonlinearities. In appendix, we study the existence properties of periodic solutions of the triple power nonlinearity.
Paper Structure (13 sections, 10 theorems, 141 equations, 13 figures)

This paper contains 13 sections, 10 theorems, 141 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Analysis of the profile equation
  3. The minimization problems
  4. A Fourier rearrangement inequality
  5. Minimization on the mass constraint
  6. The focusing case in $P_T$
  7. The defocusing case in $P_T$
  8. The defocusing case in $A_T$
  9. Minimization on the Nehari manifold
  10. The focusing case in $P_T$
  11. The defocusing case in $P_T$
  12. The focusing case in $A_T$
  13. Triple power nonlinearity

Key Result

Theorem 1.1

Let the energy, mass, momentum, action and Nehari functionals be defined as above. The following holds.

Figures (13)

  • Figure 1: Possible plots of $V$ as a function of $r$ when $J=0$.
  • Figure 2: Possible plots of $k$ as a function of $r$.
  • Figure 3: Possible plots of $V_J$ as a function of $r$ with $J \neq 0$.
  • Figure 4: Examples of phase portraits of the solutions for the defocusing case when $J=0$.
  • Figure 5: Examples of phase portraits of the solutions for the focusing case when $J=0$.
  • ...and 8 more figures

Theorems & Definitions (22)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Remark 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 12 more