Multiscale expansion on the lattice and integrability of partial difference equations

TL;DR

This paper proposes a conjecture that uses discrete multiscale analysis to test the integrability and linearizability of dispersive lattice equations, linking their properties to the resulting nonlinear or linear Schrödinger equations.

Contribution

It introduces a novel conjecture connecting lattice equation integrability with the form of derived Schrödinger equations via multiscale expansion, supported by multiple examples.

Findings

Integrable lattice equations lead to integrable NLS equations.

Linearizable lattice equations lead to linear Schrödinger equations.

Non-integrable lattice equations can produce non-integrable NLS equations.

Abstract

We conjecture an integrability and linearizability test for dispersive Z^2-lattice equations by using a discrete multiscale analysis. The lowest order secularity conditions from the multiscale expansion give a partial differential equation of the form of the nonlinear Schrodinger (NLS) equation. If the starting lattice equation is integrable then the resulting NLS equation turns out to be integrable, while if the starting equation is linearizable we get a linear Schrodinger equation. On the other hand, if we start with a non-integrable lattice equation we may obtain a non-integrable NLS equation. This conjecture is confirmed by many examples.