- 96-377 F. Gesztesy and R. Weikard
- Toward a Characterization of Elliptic Solutions of Hierarchies of
Aug 21, 96
(auto. generated ps),
of related papers
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
$q$ 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
complex value of the spectral parameter $E$.
This result also provides an explicit condition under which a
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
$n \times n$ systems of differential equations with elliptic
as coefficients associated with Gelfand-Dickey and matrix-valued
hierarchies of soliton equations.