Laminated Wave Turbulence: Generic Algorithms II

TL;DR

This paper introduces a second generic algorithm for solving Diophantine equations related to wave resonance conditions in laminated wave turbulence, addressing computational challenges for large-scale problems.

Contribution

It presents a new two-class-case generic algorithm for solving resonance Diophantine equations in laminated wave turbulence, complementing previous work on the one-class-case algorithm.

Findings

Developed a two-class-case generic algorithm for wave resonance equations.

Addressed computational complexity for equations with large integer solutions.

Extended the framework for solving Diophantine equations in wave turbulence models.

Abstract

The model of laminated wave turbulence puts forth a novel computational problem - construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order $10^{12}$ and more. The equations to be solved in integers are resonant conditions for nonlinearly interacting waves and their form is defined by the wave dispersion. It is established that for the most common dispersion as an arbitrary function of a wave-vector length two different generic algorithms are necessary: (1) one-class-case algorithm for waves interacting through scales, and (2) two-class-case algorithm for waves interacting through phases. In our previous paper we described the one-class-case generic algorithm and in our present paper we present the two-class-case generic algorithm.