Trivalent Graphs and Solitons

TL;DR

This paper demonstrates that fourth order real self-adjoint difference operators on trivalent trees can be deformed into integrable nonlinear systems using L-A-B triples, with constructed Laplace transformations, unlike second order operators.

Contribution

It introduces a novel class of integrable systems derived from fourth order difference operators on trivalent trees, expanding the understanding of discrete integrable models.

Findings

Existence of nontrivial deformations preserving energy levels

Construction of L-A-B triples for these operators

Laplace transformations for the operators

Abstract

It is shown that the fourth order real self-adjoint difference operator on the Tivalent Tree admits nontrivial deformations preserving one energy level and therefore defines a nontrinial hierarhy of the completely integrable nonlinear systems representible through the ''L-A-B-triple''. The Laplace transformations for these operators are also constructed. Nothing like that exists for the second order difference operators on this tree.