 96377 F. Gesztesy and R. Weikard
 Toward a Characterization of Elliptic Solutions of Hierarchies of
Soliton Equations
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Aug 21, 96

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Abstract. The current status of an explicit characterization of all elliptic
algebrogeometric 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 secondorder ordinary
differential equations with elliptic coefficients associated with a
common period lattice. The fundamental link between Picard's theorem
and elliptic algebrogeometric 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 $n$thorder scalar differential equations and
firstorder
$n \times n$ systems of differential equations with elliptic
functions
as coefficients associated with GelfandDickey and matrixvalued
hierarchies of soliton equations.
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