Equations of 3-th waves hierarchy

TL;DR

This paper constructs specific equations within the 3-th wave hierarchy using discrete transformations, highlighting differences from $A_1$ algebra cases, and focuses on systems of zero, first, and second degree.

Contribution

It introduces a method for constructing equations of the 3-th wave hierarchy for the $A_2$ algebra, emphasizing the existence of two distinct systems of the same degree.

Findings

Constructed equations of zero, first, and second degree for the 3-th wave hierarchy.

Identified key differences between $A_2$ and $A_1$ algebra systems.

Highlighted the complexity and current lack of a general solution method.

Abstract

By the method of discrete transformation equations of 3-th wave hierarchy are constructed. The main difference compare with the systems connected with $A_1$ algebra consists in a fact that in $A_2$ case there are two different systems of equations of the same degree (the maximal derivatives with respect to space coordinate). In the present paper we construct only system of equations of zero, first and second degree. At this moment the general method for solution of this problem (the type of canonical Hamiltonian operators) is unknown for the author and our direct calculationsare sufficiently combersome.