Symplectic forms in the theory of solitons

TL;DR

This paper develops a Hamiltonian framework for 2D soliton equations, identifying symplectic structures and connecting them with various theories like WKB and Seiberg-Witten, advancing understanding of soliton dynamics.

Contribution

It introduces a universal symplectic form for 2D soliton equations and compares it with forms in related theories, extending the Hamiltonian formalism to higher dimensions.

Findings

Identified spaces of doubly periodic operators with Hamiltonian flows

Constructed higher order symplectic forms and compared with 1D case

Connected symplectic forms in soliton theory with those in WKB, topological, and Seiberg-Witten theories

Abstract

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 $\omega={1\over 2}\r_{\infty} <\Psi_0^*\delta L\wedge\delta\Psi_0>\d k$. 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.