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Kruskal-style algorithm for cubic Schrödinger equation molecule reduction

Yvain Bruned, Valentin Clarisse

Abstract

We are interested in the molecule reduction algorithm introduced by Deng and Hani in arXiv:2104.11204. In this article, the authors use this algorithm to establish a rigidity theorem, which plays a central role in the kinetic-time derivation of the wave equation associated with the cubic Schrödinger equation. In the present article, we show that this algorithm is a graph traversal algorithm of Kruskal type, and we prove that it constructs a Kruskal spanning tree of the input molecule. This reveals the origin of the main tool for deriving kinetic equations which has also been used for the long time derivation of the Boltzmann equation.

Kruskal-style algorithm for cubic Schrödinger equation molecule reduction

Abstract

We are interested in the molecule reduction algorithm introduced by Deng and Hani in arXiv:2104.11204. In this article, the authors use this algorithm to establish a rigidity theorem, which plays a central role in the kinetic-time derivation of the wave equation associated with the cubic Schrödinger equation. In the present article, we show that this algorithm is a graph traversal algorithm of Kruskal type, and we prove that it constructs a Kruskal spanning tree of the input molecule. This reveals the origin of the main tool for deriving kinetic equations which has also been used for the long time derivation of the Boltzmann equation.
Paper Structure (6 sections, 5 theorems, 15 equations, 17 figures)

This paper contains 6 sections, 5 theorems, 15 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Trees, couples and molecules
  3. Rigidity theorem and algorithm
  4. Steps and proof of Proposition \ref{['acyclic_steps']}
  5. Algorithm and proof of Theorem \ref{['main_theorem']}
  6. Example of spanning tree

Key Result

theorem 1

For all $\beta \in (\mathbb{R}^+)^d \setminus \mathcal{N}$, $A \geqslant 40d$, $\phi_{\text{in}} \in \mathcal{S}(\mathbb{R}^d, \mathbb{R}^+)$, there exists $\delta > 0$ such that, for $L$ sufficiently large satisfying the scaling law $\lambda^2 = L^{d-1}$, the cubic Schrödinger equation admits a smo

Figures (17)

  • Figure 1: Example of Prim minimum spanning tree rooted at $A$
  • Figure 2: Example of Kruskal minimum spanning tree
  • Figure 3: Examples of a tree and a couple. Signs are displayed inside each node of the tree. Paired leaves have the same color and they come with different sign.
  • Figure 4: Type I molecular chain
  • Figure 5: Type II molecular chain
  • ...and 12 more figures

Theorems & Definitions (15)

  • theorem 1: Deng--Hani
  • theorem 2
  • proposition 1
  • definition 1: Tree and couple
  • definition 2: Decoration
  • definition 3: Molecule
  • proposition 2
  • proof
  • definition 4: Base molecule associated with a couple
  • proof
  • ...and 5 more