Systematic method of generating new integrable systems via inverse Miura maps

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.

48 pages; The method and results were implicitly used in arXiv:nlin/0105053 and arXiv:0712.4373; (v2) added one paragraph on p.12, to appear in JMP Vol.52 (2011)

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• The Algebro-Geometric Solutions for the Ruijsenaars-Toda Hierarchy
• On a new integrable discretization of the derivative nonlinear Schrödinger (Chen-Lee-Liu) equation
• Unitarily-invariant integrable systems and geometric curve flows in $SU(n+1)/U(n)$ and $SO(2n)/U(n)$
• A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices
• Functional representation of the negative DNLS hierarchy