Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete wave turbulence for a coupled system of quintic Schrödinger equations

Shayan Zahedi

Abstract

We derive rigorously the non-linear macroscopic system associated to a microscopic system of coupled quintic Schrödinger equations in the framework of discrete wave turbulence under a particular scaling law that describes the limiting process. Our system evolves from a pair of well-prepared random initial data. More precisely, in dimensions $d\geq2$, we set up our microscopic system on a large box of size $L$ with weak non-linearity of strength $ε$. In the limit $L\to\infty$ and $ε\to0$, under the scaling law $εL^{\frac{1}β}=1$ with $β\in(1,\infty)$, we prove that the long-time behaviour of our microscopic system is statistically described up to times $δε^{-1}$ by a non-linear resonant system whose dynamics are driven by exact resonances, where $δ$ is independent of $L$ and $ε$. Our system does not display generic symmetries, in particular not mass conservation. In such systems with fewer invariances, exact resonances contribute significantly compared to quasi-resonances and are essentially responsible for the effective dynamics in the large-box limit. We justify the emergence of discrete wave turbulence for our microscopic model.

Discrete wave turbulence for a coupled system of quintic Schrödinger equations

Abstract

We derive rigorously the non-linear macroscopic system associated to a microscopic system of coupled quintic Schrödinger equations in the framework of discrete wave turbulence under a particular scaling law that describes the limiting process. Our system evolves from a pair of well-prepared random initial data. More precisely, in dimensions , we set up our microscopic system on a large box of size with weak non-linearity of strength . In the limit and , under the scaling law with , we prove that the long-time behaviour of our microscopic system is statistically described up to times by a non-linear resonant system whose dynamics are driven by exact resonances, where is independent of and . Our system does not display generic symmetries, in particular not mass conservation. In such systems with fewer invariances, exact resonances contribute significantly compared to quasi-resonances and are essentially responsible for the effective dynamics in the large-box limit. We justify the emergence of discrete wave turbulence for our microscopic model.
Paper Structure (37 sections, 50 theorems, 369 equations, 2 figures)

This paper contains 37 sections, 50 theorems, 369 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Statement of the main result
  4. The microscopic system \ref{['system']}
  5. The macroscopic system \ref{['dee']}
  6. The main result
  7. Heuristic derivation of \ref{['dee']}
  8. Ingredients of the proof
  9. Feynman diagrams and couples
  10. Reformulation in terms of signed and coloured trees
  11. Decorations and averaging over rooted trees
  12. First part of the proof of \ref{['this is the main theorem']}
  13. Initial estimates and preparational tools
  14. Passing from sums to integrals
  15. Weighing gains against losses
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $d\geq2$, $s>\frac{d}{2}$ and $\beta\in(1,\infty)$. There exist $\delta,L_0,A_0>0$ such that for all $L\geq L_0$ and $A\geq A_0$:

Figures (2)

  • Figure 2.1: Convention for the nonlinearity in \ref{['system']}
  • Figure 5.1: These are the first six cases of \ref{['need to code this']}, where the resonant branching occurs for the left tree. The remaining six cases can be drawn by switching the left with the right tree and exchanging the sign $\iota\leftrightarrow-\iota$ and the colour $\eta\leftrightarrow\eta'$.

Theorems & Definitions (142)

  • Remark 1
  • Theorem 1.1
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 1
  • Remark 5
  • Definition 2
  • Remark 6
  • Definition 3
  • ...and 132 more