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Energy transfer for solutions to the nonlinear Schrödinger equation on irrational tori

Alexander Hrabski, Yulin Pan, Gigliola Staffilani, Bobby Wilson

Abstract

We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schrödinger (NLS) initial value problem on 2D irrational tori. Moreover we complement the analytic study with numerical experimentation. As a biproduct of our investigation we also prove that the quasi-resonant part of the NLS initial value problem we consider, in both the focusing and defocusing case, is globally well-posed for initial data of finite mass.

Energy transfer for solutions to the nonlinear Schrödinger equation on irrational tori

Abstract

We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schrödinger (NLS) initial value problem on 2D irrational tori. Moreover we complement the analytic study with numerical experimentation. As a biproduct of our investigation we also prove that the quasi-resonant part of the NLS initial value problem we consider, in both the focusing and defocusing case, is globally well-posed for initial data of finite mass.

Paper Structure

This paper contains 12 sections, 7 theorems, 66 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. $\ell^2$ Sobolev Spaces
  4. The Quasi-resonant Sets
  5. The Quasi-resonance Condition and the Irrationality of the Torus
  6. Dynamics of the Quasi-resonant System
  7. Numerical Study
  8. Numerical Setup
  9. Results
  10. Kinematic Study on $(\Lambda,\tau)$-quasi Resonant Sets
  11. Discussions on Numerical Study
  12. On the Finite-precision Representation of Irrational $\omega^2$

Key Result

theorem 1

Assume that the torus $\mathbb{T}^2_{\underline{\omega}}$ is a generic irrational ordered pair. Consider the quasi-resonant focusing and defocusing NLS initial value problem in which $(|\psi|^2\psi)^*$ indicates that the nonlinear interactions are only taking place on a quasi-resonantThe definition of a quasi-resonant set is in Definition quasi-res. set. Then irrNLS-res is globally well posed for

Figures (6)

  • Figure 1: The relationship between $M$ and $R$ for different $\varepsilon$ on log-scale axes for $s=2$ and $T=20T_f$. Data for $\omega^2=\sqrt{2}$ (-----) and $\omega^2=1$ (-- -- --) are included, with lines showing the median value and uncertainty bars showing the maximum and minimum of $M$ among the 5 simulations.
  • Figure 2: The 2D energy spectra of (a) the initial condition, (b) the irrational torus at $t=20T_f$ and (c) the rational torus at $t=20T_f$. Note that (b) and (c) share the color bar. The zero mode has amplitude $0$, but is colored for simplicity.
  • Figure 3: The growth of $\|\hat{\psi}(t)\|_s$ with 5 realizations of $\phi_k$
  • Figure 4: The long-time growth of $\|\hat{\psi}(t)\|_s$ on the irrational torus for $R=3.6526,2.5828,1.8263$ from top to bottom.For each $R$, the growth with $5$ realizations of $\phi_k$ are provided.
  • Figure 5: levels of $(\Lambda,\tau)$-quasi resonant sets for $L_1=[-2,2]\times[-2,2]$, $\tau=0.1$, computed up to level $N=min(6,\mathcal{N})$, where $\mathcal{N}$ corresponds to the level for which no modes can be excited in the next level. Left column: $\Lambda=10,20,30$ on irrational torus from top to bottom; Right column: $\Lambda=10,20,30$ on rational torus from top to bottom.
  • ...and 1 more figures

Theorems & Definitions (18)

  • theorem 1
  • remark 1
  • theorem 2
  • definition 1
  • definition 2
  • theorem 3
  • corollary 1
  • proof
  • lemma 1
  • proof
  • ...and 8 more