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An extension of the algebraic-geometric method for nonlinear integrable PDE'sis 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 ofthese nonlinear PDE's, which are associated to nonlinear subvarieties ofhyperelliptic Jacobians. The main results of the present paper are twofold.First, we exhibit some of the special features of integrable PDE's that admitpiecewise smooth weak solutions, which make them different from equations whosesolutions are globally meromorphic, such as the KdV equation. Second, we blendthe techniques of algebraic geometry and weak solutions of PDE's to gainfurther insight into, and explicit formulas for, piecewise-smooth finite-gapsolutions.Comment: 31 pages, no figures, to appear in Commun. Math. Phy

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type