2001/05/09 by Mark Alber, Mark S. Alber, Roberto Camassa +3 · 7 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #nlin.CD #nlin.SI
paper · pdf · doi:10.1007/pl00005573
31 pages, no figures, to appear in Commun. Math. Phys
arxiv created 2001/05/09 · openalex publication_date 2001/07/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.