Fay's identity on Riemann surfaces is a powerful tool in the context of algebro-geometric solutions to integrable equations. This relation generalizes a well-known identity for the cross-ratio function in the complex plane. It allows to establish relations between theta functions and their derivatives. This offers a complementary approach to algebro-geometric solutions of integrable equations with certain advantages with respect to the use of Baker-Akhiezer functions. It has been successfully applied by Mumford et al. to the Korteweg-de Vries, Kadomtsev-Petviashvili and sine-Gordon equations. Following this approach, we construct algebro-geometric solutions to the Camassa-Holm and Dym type equations, as well as solutions to the multi-component nonlinear Schrödinger equation and the Davey-Stewartson equations. Solitonic limits of these solutions are investigated when the genus of the associated Riemann surface drops to zero. Moreover, we present a numerical evaluation of algebro-geometric solutions of integrable equations when the associated Riemann surface is real.
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-00622289 |
| Date | 27 June 2011 |
| Creators | Kalla, Caroline |
| Publisher | Université de Bourgogne |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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