 07150 F. Gesztesy, H. Holden, J. Michor, and G. Teschl
 The algebrogeometric initial value problem for the AblowitzLadik hierarchy
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Jun 22, 07

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Abstract. We discuss the algebrogeometric initial value problem for the AblowitzLadik 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 AblowitzLadik hierarchy, which is of independent interest as it solves the inverse
algebrogeometric spectral problem for generally nonunitary AblowitzLadik Lax 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 Dubrovintype equations), this yields the construction of global algebrogeometric solutions of the timedependent AblowitzLadik hierarchy.
The generally nonunitary behavior of the underlying Lax operator associated with general coefficients for the AblowitzLadik 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 AblowitzLadik hierarchy but applies generally to 1+1dimensional completely integrable (discrete and continuous) soliton equations.
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