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2010 Équations couplées non-linéaires de Schrödinger
A. Lesfari
Afr. Diaspora J. Math. (N.S.) 10(2): 96-108 (2010).


In this paper, we give a complete description of the invariant surfaces of the system governing the motion of the coupled nonlinear Schrödinger equations and their completion into abelian surfaces. We derive the associated Riemann surface on the basis of Painlevé-type analysis in the form of a genus $3$ Riemann surface $\Gamma $, which is a double ramified covering of an elliptic curve $\Gamma _{0}$ and a two sheeted genus two hyperelliptic Riemann surface $C$. We show that the affine surface $V_{c}$ obtained by setting the two quartics invariants of the problem equal to generic constants, is the affine part of an abelian surface $\widetilde{V}_{c}.$ The latter can be identified as the dual of the Prym variety ${\rm Pr} ym (\Gamma /\Gamma _{0})$ on which the problem linearizes, that is to say their solutions can be expressed in terms of abelian integrals. Also, we discuss a connection between $\widetilde{V}_{c}$ and the jacobian variety $Jac(C)$ of the genus $2$ hyperelliptic Riemann surface $C$.


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A. Lesfari. "Équations couplées non-linéaires de Schrödinger." Afr. Diaspora J. Math. (N.S.) 10 (2) 96 - 108, 2010.


Published: 2010
First available in Project Euclid: 29 November 2010

zbMATH: 1263.14045
MathSciNet: MR2774262

Primary: 14K35
Secondary: 14M10, 37J35

Rights: Copyright © 2010 Mathematical Research Publishers


Vol.10 • No. 2 • 2010
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