On the numerical evaluation of algebro-geometric solutions to integrable equations (Q2882401)

The authors address typical analytical problems appearing in the numerical study of theta-functional solutions to integrable PDEs and present the state of the art of the field by considering concrete examples. They first recall some basic facts from the theory of multi-dimensional theta functions and the theory of real Riemann surfaces, necessary to give theta-functional solutions to the n-NLS (nonlinear Schrödinger) and DS (Davey-Stewartson) equations. Then, the authors consider the hyperelliptic case and study concrete examples of low genus, also in almost degenerate situations. They also consider examples of non-hyperelliptic real Riemann surfaces and discuss symplectic transformations needed to obtain smooth solutions. The paper concludes with some remarks.