96-377 F. Gesztesy and R. Weikard

Toward a Characterization of Elliptic Solutions of Hierarchies of Soliton Equations (100K, amslatex) Aug 21, 96

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Abstract. The current status of an explicit characterization of all elliptic algebro-geometric solutions of hierarchies of soliton equations is discussed and the case of the KdV hierarchy is considered in detail. More precisely, we review our recent result that an elliptic function $q$ is a solution of some equation of the stationary KdV hierarchy, if and only if the associated differential equation $\psi''(E,z)+ q(z)\psi(E,z)=E\psi(E,z)$ has a meromorphic fundamental system for every complex value of the spectral parameter $E$. This result also provides an explicit condition under which a classical theorem of Picard holds. This theorem guarantees the existence of solutions which are elliptic of the second kind for second-order ordinary differential equations with elliptic coefficients associated with a common period lattice. The fundamental link between Picard's theorem and elliptic algebro-geometric solutions of completely integrable hierarchies of nonlinear evolution equation is the principal new aspect of our approach. In addition, we describe most recent attempts to extend this circle of ideas to $n$-th-order scalar differential equations and first-order $n \times n$ systems of differential equations with elliptic functions as coefficients associated with Gelfand-Dickey and matrix-valued hierarchies of soliton equations.

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