Fine structure of the discrete transformation for multicomponent integrable systems

TL;DR

This paper reveals that for multicomponent integrable systems related to algebras $A_n$, the discrete transformation can be decomposed into commuting basis transformations, with detailed analysis for a 3-wave system.

Contribution

It demonstrates the fine structure of the discrete transformation in multicomponent integrable systems and provides explicit calculations for the 3-wave case.

Findings

Discrete transformation $T$ decomposes into commuting basis transformations $T_i$.

The structure is explicitly detailed for the 3-wave interacting system.

The decomposition allows for a better understanding of the transformation's properties.

Abstract

It is shown that in the case of multicomponent integrable systems connected with algebras $A_n$, the discrete transformation $T$ possesses the fine structure and can be represented in the form $T=\prod T_i^{l_i}$, where $T_i$ are n commuting basis discrete transformations, and $l_i$ are arbitrary natural numbers. All the calculations are conducted in detail for the case of a 3-wave interacting system.