Multiscale expansion on the lattice and integrability of partial difference equations

2007/10/28 by Rafael Hernández Heredero, Heredero, Rafael Hernandez, D. Levi +5
Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.0710.5299

openalex publication_date 2007/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We conjecture an integrability and linearizability test for dispersive Z²-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.

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