Integrability test of discrete nonlinear Schrödinger equations via multiscale reduction

Abstract

In this article, we consider the multiscale reduction around the harmonic solution of a general discrete nonlinear Schrödinger equation (dNLSE) depending on constant coefficients. According to the values of the coefficients we can have both integrable and non-integrable dNLSEs. For all values of the coefficients entering the dNLSE, non-secularity conditions provide an integrable NLSE at the lowest order in the perturbation parameter. However at higher order in the perturbation expansion the request that the expansion is compatible with the NLSE hierarchy gives integrability conditions which are not satisfied for the non-integrable dNLSEs.

Keywords:

• discrete equations
• nonlinear Schrödinger equation
• multiscale reduction technique
• integrability test

AMS Subject Classifications::

• 17B80
• 34E13
• 35R10
• 37K10
• 39-XX