The String Equation and Solitons

TL;DR

This paper surveys the theory of solitons with a focus on periodic solutions, algebraic geometry, and the string equation, including new results on 2D periodic Schrödinger operators and Laplace transformations.

Contribution

It introduces new findings on 2D periodic Schrödinger operators and cyclic Laplace transformations within the context of soliton theory and algebraic geometry.

Findings

New results on 2D periodic Schrödinger operators.

Analysis of cyclic and semicyclic Laplace transformations.

Connections between solitons, algebraic geometry, and the string equation.

Abstract

We present a brief survey of the results of the Theory of Solitons from the viewpoint of the periodic theory including some new results in the theory of 2-dimensional periodic Schrodinger Operators. The main subjects are: Periodic Solitons and Algebraic Geometry, The Theory of Solitons and the String equation, Topologically trivial and nontrivial periodic two-dimensional Schrodinger operators and Riemann surfaces and Cyclic and semicyclic chains of Laplace transformations (new results of the present author and A.Veselov).