1997/08/29 by I. M. Krichever, Krichever, I. M., D. H. Phong +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9708170
76 pages, Tex, no figures
arxiv created 1997/08/29 · openalex publication_date 1997/08/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form ω=1\over 2\r∞ \d k. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finite-gap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in non-linear WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.