Cauchy problem for integrable discrete equations on quad-graphs

TL;DR

This paper investigates initial value problems for integrable discrete equations on quad-graphs, establishing a geometric criterion for well-posedness and exploring solution interactions with defects, including examples of solitons and kinks.

Contribution

It introduces a geometric criterion for well-posedness of discrete integrable equations on quad-graphs and analyzes solution interactions with defects.

Findings

Established a geometric criterion for well-posedness.

Analyzed interactions of solutions with localized defects.

Presented examples of solitons and kinks on various quad-graphs.

Abstract

Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects in the regular square lattice are discussed for the discrete potential KdV and linear wave equations. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.