Integrable Systems and Geometry of Riemann Surfaces

TL;DR

This paper explores the deep connections between integrable systems and Riemann surface geometry, providing explicit solutions, geometric interpretations, and extending analysis to the nonlinear Schrödinger equation.

Contribution

It offers a comprehensive, self-contained overview of integrable systems' relation to Riemann surfaces, including explicit solutions, geometric insights, and new results for NLS equations.

Findings

Explicit polynomial solutions for stationary KdV at small degrees

Formula for hyper-elliptic curves parametrizing N-solitons

Analysis of time evolution in KdV and NLS equations

Abstract

We give a self-contained introduction to the relations between Integrable Systems and the Geometry of Riemann Surfaces. We start from a historical introduction to the topic of integrable systems. Afterwards, we study the polynomial solutions to the stationary KdV equation, giving concrete examples of the computations of the solutions on small degree ($N=0$, $1$, $2$.) We discused the geometry of the so called "square eigen-functions" for those cases. Later, we present the solutions in the general case, presenting a formula for the hyper-elliptic curve of genus $N$ that parametrizes the solutions, the $N$-solitons. We relate also our equations to the Lax hierarchy, and analyze the time evolution for the KdV. Using the scalar operator for the NLS equation, computed by Kamchatnov, Krankel and Umarov, we analyse such an equation, following a similar approach as the one we used for the KdV equation. We present some original results obtained in that case.