Quadratic invariants for discrete clusters of weakly interacting waves

TL;DR

This paper develops a linear algebra framework to identify quadratic invariants in discrete clusters of weakly interacting waves, linking topological cluster properties to conservation laws, with applications to wave models.

Contribution

It introduces an algorithm to compute independent quadratic invariants from the cluster topology, reducing the problem to linear algebra and classifying small clusters.

Findings

Number of invariants equals N - M*

Invariants relate to cluster topology and linkage

Algorithm decomposes large clusters into smaller ones

Abstract

We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular (M times N) matrix A with entries 1, -1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J = N - M* (greater than or equal to N - M), where M* is the number of linearly independent rows in A. Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including the basic structures leading to M* < M. We illustrate our findings by presenting examples from the Charney-Hasegawa-Mima wave model, and by showing a classification of small (up to three-triad) clusters.