Inverse scattering method for Hitchin systems of types $B_n$, $C_n$, $D_n$, and their generalizations

TL;DR

This paper develops an inverse scattering method tailored for Hitchin systems of types B, C, D, extending classical approaches to systems with semi-simple group symmetries on Riemann surfaces.

Contribution

It introduces a novel inverse scattering framework specifically for Hitchin systems with semi-simple group symmetries, broadening the scope beyond traditional $GL(n)$ cases.

Findings

Solution of the inverse scattering problem for Hitchin systems of types B, C, D.

Extension of inverse scattering techniques to systems with semi-simple group symmetry.

Applicable to integrable systems with spectral parameter on Riemann surfaces.

Abstract

We give a solution of the Inverse Scattering Problem for integrable systems with a finite number degrees of freedom, admitting a Lax representation with spectral parameter on a Riemann surface. While conventional approaches deal with the systems with $GL(n)$ symmetry, we focus on the problems arising in the case of symmetry with respect to a semi-simple group. Our main results apply to Hitchin systems of the types $B_n$, $C_n$, $D_n$.