 0615 Fritz Gesztesy, Helge Holden, and Gerald Teschl
 The AlgebroGeometric Toda Hierarchy Initial Value Problem for ComplexValued Initial Data
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Jan 18, 06

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Abstract. We discuss the algebrogeometric initial value problem for the Toda hierarchy with complexvalued 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 complexvalued algebrogeometric solutions of the Toda hierarchy, which is of independent interest as it solves the inverse algebrogeometric spectral problem for generally nonselfadjoint Jacobi operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate firstorder system of differential equations with respect to time (a substitute for the wellknown Dubrovin equations), this yields the construction of global algebrogeometric solutions of the timedependent Toda hierarchy.
The inherent nonselfadjointness of the underlying Lax (i.e., Jacobi) operator associated with complexvalued 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+1dimensional completely integrable (discrete and continuous) soliton equations.
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