Revista Matemática Iberoamericana

Full-Text PDF (476 KB) | Metadata | Table of Contents | RMI summary
Volume 24, Issue 1, 2008, pp. 117–182
DOI: 10.4171/RMI/532

Published online: 2008-04-30

The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data

Fritz Gesztesy[1], Helge Holden[2] and Gerald Teschl[3]

(1) Baylor University, Waco, USA
(2) University of Trondheim, Norway
(3) Universität Wien, Austria

We discuss the algebro-geometric initial value problem for the Toda hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Toda hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for generally non-self-adjoint Jacobi operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin equations), this yields the construction of global algebro-geometric solutions of the time-dependent Toda hierarchy. The inherent non-self-adjointness of the underlying Lax (i.e., Jacobi) operator associated with complex-valued coefficients for the Toda hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Toda hierarchy but applies generally to $1+1$-dimensional completely integrable discrete soliton equations.

Keywords: Toda hierarchy, complex-valued solutions, initial value problem

Gesztesy Fritz, Holden Helge, Teschl Gerald: The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data. Rev. Mat. Iberoam. 24 (2008), 117-182. doi: 10.4171/RMI/532