Unique continuation for discrete nonlinear wave equations

TL;DR

This paper proves unique continuation properties for several discrete nonlinear wave equations, showing that solutions coinciding at a single point over a short time imply they are identical everywhere, regardless of integrability.

Contribution

It demonstrates unique continuation for discrete nonlinear wave equations without relying on integrability, applicable to multiple hierarchies.

Findings

Unique continuation holds for Toda lattice, Kac-van Moerbeke, and Ablowitz-Ladik equations.

Proof does not depend on integrability, allowing broader application.

Solutions coinciding at one point over a short time are identical everywhere.

Abstract

We establish unique continuation for various discrete nonlinear wave equations. For example, we show that if two solutions of the Toda lattice coincide for one lattice point in some arbitrarily small time interval, then they coincide everywhere. Moreover, we establish analogous results for the Toda, Kac-van Moerbeke, and Ablowitz-Ladik hierarchies. Although all these equations are integrable, the proof does not use integrability and can be adapted to other equations as well.