Systematic method of generating new integrable systems via inverse Miura maps

TL;DR

This paper introduces a systematic method to generate new integrable systems by interpreting the Lax representation as a linearized Miura transformation, enabling the construction of modified systems via inverse Miura maps across various dimensions.

Contribution

It provides a novel interpretation of the Lax representation and a systematic approach to identify and generate new integrable systems through inverse Miura transformations.

Findings

Successfully applied to continuous and discrete systems in 1+1 and 2+1 dimensions.

Generated new integrable systems from known ones, including NLS and Zakharov-Ito.

Demonstrated effectiveness with multiple examples.

Abstract

We provide a new natural interpretation of the Lax representation for an integrable system; that is, the spectral problem is the linearized form of a Miura transformation between the original system and a modified version of it. On the basis of this interpretation, we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by Miura transformations. Thus, this method can be used to generate new integrable systems from known systems through inverse Miura maps; it can be applied to both continuous and discrete systems in 1+1 dimensions as well as in 2+1 dimensions. The effectiveness of the method is illustrated using examples such as the nonlinear Schroedinger (NLS) system, the Zakharov-Ito system (two-component KdV), the three-wave interaction system, the Yajima-Oikawa system, the Ablowitz-Ladik lattice (integrable space-discrete NLS), and two (2+1)-dimensional NLS systems.