Laminated Wave Turbulence: Generic Algorithms III

TL;DR

This paper introduces a new generic algorithm for solving Diophantine equations related to laminated wave turbulence with rational dispersion functions, focusing on the discrete energy transport layer.

Contribution

It presents a novel algorithm specifically designed for rational dispersion functions, addressing computational challenges in wave turbulence modeling.

Findings

Algorithm effectively solves high-degree Diophantine equations in multiple variables.

Discrete layer dominates energy transport in systems with rational dispersion.

Enhanced understanding of wave turbulence dynamics without statistical layer.

Abstract

Model of laminated wave turbulence allows to study statistical and discrete layers of turbulence in the frame of the same model. Statistical layer is described by Zakharov-Kolmogorov energy spectra in the case of irrational enough dispersion function. Discrete layer is covered by some system(s) of Diophantine equations while their form is determined by wave dispersion function. This presents a very special computational challenge - to solve Diophantine equations in many variables, usually 6 to 8, in high degrees, say 16, in integers of order $10^{16}$ and more. Generic algorithms for solving this problem in the case of {\it irrational} dispersion function have been presented in our previous papers. In this paper we present a new generic algorithm for the case of {\it rational} dispersion functions. Special importance of this case is due to the fact that in wave systems with rational dispersion the statistical layer does not exist and the general energy transport is governed by the discrete layer alone.