The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy
| Autor: | Gesztesy, F., Holden, H., Michor, J., Teschl, G. |
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| Rok vydání: | 2007 |
| Předmět: | |
| Zdroj: | Discrete Contin. Dyn. Syst. 26:1, 151-196 (2010) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.3934/dcds.2009.26.151 |
| Popis: | 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. Comment: 47 pages |
| Databáze: | arXiv |
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