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Inhomogeneous turbulence for the Wick nonlinear Schrödinger equation

Zaher Hani, Jalal Shatah, Hui Zhu

Abstract

We introduce a simplified model for wave turbulence theory -- the Wick NLS, of which the main feature is the absence of all self-interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture.

Inhomogeneous turbulence for the Wick nonlinear Schrödinger equation

Abstract

We introduce a simplified model for wave turbulence theory -- the Wick NLS, of which the main feature is the absence of all self-interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture.
Paper Structure (48 sections, 56 theorems, 293 equations, 4 figures, 3 tables)

This paper contains 48 sections, 56 theorems, 293 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Wick NLS
  3. Wick product
  4. Perturbative expansion
  5. Wick renormalization
  6. Ensemble of wavepackets
  7. Effective dynamics
  8. Scaling regimes
  9. Space homogeneity
  10. Wave kinetic equation
  11. Finer effective dynamics
  12. Trees and couples
  13. Ternary trees
  14. Couples
  15. Conjugate nodes and irreducible couples
  16. ...and 33 more sections

Key Result

Theorem 1.1

For all $d \ge 1$, $\lambda > 0$, $L > 0$, $\epsilon \ge 0$, and $\phi$ is a Schwartz function on $\mathbb{R}^{2d}$, there exists a unique sequence of stochastic processes $(A_k : \mathbb{R} \times \mathbb{R}^d \to \mathfrak{H})_{k\in\mathbb{Z}^d_L}$ such that Moreover $A_k^{2n} = 0$ for all $k \in \mathbb{Z}^d_L$ and $n \in \mathbb{N}$; if $(n,k) \ne (n',k')$, then for all $t,t' \in \mathbb{R}$

Figures (4)

  • Figure 1.1: Wave turbulence theory for WNLS
  • Figure 2.1: Tree planting
  • Figure 2.2: Examples of couple factorization
  • Figure 2.3: An example of $\mathscr{A}_{\mathfrak{p}_1} \cap \mathscr{A}_{\mathfrak{p}_2}$

Theorems & Definitions (124)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: New trees from old
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 114 more