New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation

Abstract

An independent derivation of solutions to the Camassa-Holm equation in terms of multi-dimensional theta functions is presented using an approach based on Fay's identities. Reality and smoothness conditions are studied for these solutions from the point of view of the topology of the underlying real hyperelliptic surface. The solutions are studied numerically for concrete examples, also in the limit where the surface degenerates to the Riemann sphere, and where solitons and cuspons appear.

19 pages, 9 figures

References in corpus (4)

• A 2-Component Generalization of the Camassa-Holm Equation and Its Solutions
• The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's of Shallow Water and Dym Type
• Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP solutions
• On the numerical evaluation of algebro-geometric solutions to integrable equations

Cited by in corpus (3)

• Explicit solutions for nonlocal NLS: GBDT and algebro-geometric approaches
• Computational approach to hyperelliptic Riemann surfaces
• Higher order degenerations of Fay's identities and applications to integrable equations