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.
Keywords
Cite
@article{arxiv.1012.2458,
title = {Systematic method of generating new integrable systems via inverse Miura maps},
author = {Takayuki Tsuchida},
journal= {arXiv preprint arXiv:1012.2458},
year = {2011}
}
Comments
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)