On the numerical evaluation of algebro-geometric solutions to integrable equations

TL;DR

This paper investigates numerical methods for evaluating algebro-geometric solutions to integrable equations, focusing on hyperelliptic and general real Riemann surfaces, and discusses algorithms, basis transformations, and applications to specific equations.

Contribution

It develops efficient algorithms for numerical evaluation of solutions on hyperelliptic surfaces and studies symplectic transformations for general real Riemann surfaces, including explicit formulas for M-curves.

Findings

Efficient algorithms for hyperelliptic surfaces enable solitonic limit analysis.

Symplectic transformations help adapt homology bases for general Riemann surfaces.

Applications to Davey-Stewartson and multi-component nonlinear Schrödinger equations.

Abstract

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solutions of the Davey-Stewartson and the multi-component nonlinear Schr\"odinger equations.