Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2001-05-09
Nonlinear Sciences
Chaotic Dynamics
31 pages, no figures, to appear in Commun. Math. Phys
Scientific paper
10.1007/PL00005573
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.
Alber Mark S.
Camassa Roberto
Fedorov Yuri N.
Holm Darryl D.
Marsden Jerrold E.
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