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

29 pages, 20 figures

References in corpus (1)

• Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP solutions

Cited by in corpus (4)

• Numerical calculation of N-periodic wave solutions to coupled KdV-type equations
• High precision numerical approach for the Davey-Stewartson II equation for Schwartz class initial data
• Data Transmission based on Exact Inverse Periodic Nonlinear Fourier Transform, Part I: Theory
• New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation