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Geometric study of a family of integrable systems. (Étude géométrique d’une famille de systémes intégrables.) (French) Zbl 1133.37330

Summary: In this paper, we consider a hierarchy of Hamiltonian systems. Usually, this system is nonintegrable, but we give two integrable cases in the sense of Liouville. In the first case, we show that the system is linearized in the Jacobian variety of a smooth hyperelliptic Riemann surface. For the second case, we describe a connection with the system of two coupled nonlinear Schrödinger equations. We use this connection for deriving a Lax representation and a spectral curve for the system. The linearized flow can be realized on an elliptic curve.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
14H70 Relationships between algebraic curves and integrable systems
70G55 Algebraic geometry methods for problems in mechanics
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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