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January  2010, 26(1): 151-196. doi: 10.3934/dcds.2010.26.151

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

Fritz Gesztesy 1, , Helge Holden 2, , Johanna Michor 3, and  Gerald Teschl 3,

1. 

Department of Mathematics, University of Missouri, Columbia, MO 65211, United States
2. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim
3. 

Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria, Austria

Received  August 2008 Revised  June 2009 Published  October 2009

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik 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 Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax 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-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy.
   The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik 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 Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.
Keywords: initial value problem., complex-valued solutions, Ablowitz-Ladik hierarchy.
Mathematics Subject Classification: Primary 37K10, 37K20, 47B36; Secondary 35Q58, 37K6.
Citation: Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151
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