Fay's identity in the theory of integrable systems

Abstract : 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.
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https://tel.archives-ouvertes.fr/tel-00622289
Contributor : Caroline Kalla <>
Submitted on : Monday, September 12, 2011 - 1:26:50 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Tuesday, December 13, 2011 - 2:22:24 AM

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  • HAL Id : tel-00622289, version 1

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Caroline Kalla. Fay's identity in the theory of integrable systems. Mathematics [math]. Université de Bourgogne, 2011. English. ⟨tel-00622289v1⟩

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