Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs

TL;DR

This paper explores how discrete nonlinear equations, including solitons and Lax pairs, can be extended from one-dimensional systems to Z-graded graphs, revealing conditions for such lifts and their limitations.

Contribution

It introduces a method for lifting 1D nonlinear equations to Z-graded graphs and proves the existence of solitons on graded fractal graphs, highlighting new insights into graph-based integrable systems.

Findings

Solitons exist on graded fractal graphs with static potentials.

Lax pairs do not always lift to Z-graded graphs, even in simple cases.

Conditions for lifting solutions to graphs are identified.

Abstract

We consider discrete one dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons {for static potentials} on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.