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Analysis of quasi-periodic waves of cubic nonlinear Schr{ö}dinger equations

Perla Kfoury, Stefan Le Coz, Tai-Peng Tsai

TL;DR

The paper investigates quasi-periodic standing waves of the 1D cubic NLS in both focusing and defocusing regimes by linking the profile ODE invariants $(J,E)$ to the NLS invariants $(M,P,\mathcal{E})$. It establishes a diffeomorphic correspondence between $(J,E)$ and $(\tilde{M},\tilde{P})$ in the defocusing case, and develops a gradient-flow method with discrete normalization to compute energy minimizers under simultaneous mass and momentum constraints. Jacobi elliptic functions are used to describe ODE solutions, and a comprehensive numerical framework is built to recover ODE profiles as energy minimizers. Numerical experiments confirm that the computed minimizers coincide with the corresponding ODE solutions across parameter regimes, validating the two-constraint normalization scheme and its applicability to quasi-periodic NLS waves.

Abstract

We study the quasi-periodic standing wave solutions of the focusing and defocusing cubic nonlinear Schr{ö}dinger equations in dimension one. In the defocusing case, we establish a diffeomorphic correspondence between the invariants of the ordinary differential equation of the wave profiles and the conserved quantities of the evolution equation. We introduce a numerical scheme to compute the minimizers of the energy at fixed mass and momentum for both focusing and defocusing cases. The scheme is based on a gradient flow approach with discrete renormalization at each time step. The novelty of our scheme is that the renormalization step deals at the same time with the mass and the momentum constraints. In numerical experiments, we observe that a given solution of the profile ordinary differential equation is also a minimizer of the energy at corresponding mass and momentum.

Analysis of quasi-periodic waves of cubic nonlinear Schr{ö}dinger equations

TL;DR

The paper investigates quasi-periodic standing waves of the 1D cubic NLS in both focusing and defocusing regimes by linking the profile ODE invariants to the NLS invariants . It establishes a diffeomorphic correspondence between and in the defocusing case, and develops a gradient-flow method with discrete normalization to compute energy minimizers under simultaneous mass and momentum constraints. Jacobi elliptic functions are used to describe ODE solutions, and a comprehensive numerical framework is built to recover ODE profiles as energy minimizers. Numerical experiments confirm that the computed minimizers coincide with the corresponding ODE solutions across parameter regimes, validating the two-constraint normalization scheme and its applicability to quasi-periodic NLS waves.

Abstract

We study the quasi-periodic standing wave solutions of the focusing and defocusing cubic nonlinear Schr{ö}dinger equations in dimension one. In the defocusing case, we establish a diffeomorphic correspondence between the invariants of the ordinary differential equation of the wave profiles and the conserved quantities of the evolution equation. We introduce a numerical scheme to compute the minimizers of the energy at fixed mass and momentum for both focusing and defocusing cases. The scheme is based on a gradient flow approach with discrete renormalization at each time step. The novelty of our scheme is that the renormalization step deals at the same time with the mass and the momentum constraints. In numerical experiments, we observe that a given solution of the profile ordinary differential equation is also a minimizer of the energy at corresponding mass and momentum.
Paper Structure (16 sections, 6 theorems, 96 equations, 8 figures)

This paper contains 16 sections, 6 theorems, 96 equations, 8 figures.

Table of Contents

  1. Introduction
  2. From the ordinary differential equation to the minimization problem
  3. Jacobi elliptic functions
  4. Analysis of the profile equation
  5. The case $J \neq 0$
  6. The case $J=0$
  7. Existence domains
  8. The period function
  9. Mass and momentum inside the domain and on the boundaries
  10. Diffeomorphism
  11. From the minimization problem to the ordinary differential equation
  12. Variational problems
  13. Numerical solutions of the ordinary differential equation
  14. Continuous gradient flow with discrete normalization
  15. Discretization
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $b<0$, $a>0$ ($b>0$, $a\geqslant0$, or $b>0$, $a<0$ respectively). Assume that $(J,E) \in D_1$ ($D_2, D_3$ respectively). Let $u$ be a solution of eq:ode corresponding to $(J,E)$ and denote by $T(J,E)$ the fundamental period of $u$, which is defined in eq:period-T. We denote by $0 < y_1 < y_2 < where $S(\phi)= y_1 \cos^2 \phi + y_2 \sin^2 \phi$.

Figures (8)

  • Figure 1: Admissible domains for $(J,E)$. From left to right $(b,a)=(-1, 1)$, $(1, 1)$ and $(1, -1)$. The green lines are the graphs of $E_-$ and the orange line the graph of $E_+$.
  • Figure 2: The curve $Q\mapsto ( M_{\partial}(Q), P_{\partial}(Q))$ for, at the left $\left(b=-1, a=1,Q=\sqrt{\frac{a}{3}},\dots,\sqrt{a}\right)$, at the middle $\left(b=1, a=1,Q=\sqrt{a},\dots,1.5\right)$, and at the right $\left(b=1, a=-1,Q=0,\dots,1.5\right)$. On the right the range $Q\in\left(0, \sqrt{\frac{a}{3}}\right)$ is in red, the range $Q\in\left(\sqrt{\frac{a}{3}},1.5\right)$ is in blue, and we have added in orange the curve $(\tilde{M}(0,E), \tilde{P}(0,E))$ for $J=0$ and $E=-\frac{a^2}{4b},\dots,12$.
  • Figure 3: Comparison of the $\mathop{\mathrm{dn}}\nolimits$ function and the solutions of the ODE and the minimization problem.
  • Figure 4: Comparison of the $\mathop{\mathrm{cn}}\nolimits$ function and the solutions of the ODE and the minimization problem.
  • Figure 5: Comparison of the $\mathop{\mathrm{sn}}\nolimits$ function and the solutions of the ODE and the minimization problem.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • proof : Proof of Proposition \ref{['prop:period-T-on-E-']}
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 4 more