Équations couplées non-linéaires de Schrödinger

Abstract

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 $C$.