Kruskal-style algorithm for cubic Schr\"odinger equation molecule reduction

TL;DR

This paper demonstrates that a molecule reduction algorithm used in cubic Schrödinger equation analysis is fundamentally a Kruskal-type graph traversal, constructing a spanning tree crucial for kinetic equation derivations.

Contribution

It reveals the graph-theoretic nature of the molecule reduction algorithm, connecting it to Kruskal's algorithm and clarifying its role in kinetic equation derivations.

Findings

The algorithm is a Kruskal-type graph traversal.

It constructs a Kruskal spanning tree of the input molecule.

This explains its use in deriving kinetic equations.

Abstract

We are interested in the molecule reduction algorithm introduced by Deng and Hani. They 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 Schr\"odinger 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.