Wave Solutions of Evolution Equations and Hamiltonian Flows on Nonlinear Subvarieties of Generalized Jacobians

Abstract

The algebraic-geometric approach is extended to study solutions of N-component systems associated with the energy dependent Schrodinger operators having potentials with poles in the spectral parameter, in connection with Hamiltonian flows on nonlinear subvariaties of Jacobi varieties. The systems under study include the shallow water equation and Dym type equation. The classes of solutions are described in terms of theta-functions and their singular limits by using new parameterizations. A qualitative description of real valued solutions is provided.

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• Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy
• The Camassa-Holm hierarchy, related N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold
• Real-Valued Algebro-Geometric Solutions of the Camassa--Holm hierarchy
• Algebro-Geometric Solutions of the Camassa--Holm hierarchy
• Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions
• An alternative approach to the quasi-Periodic solutions of the Hunter-Saxton hierarchy
• An exactly solvable problem of wave fronts and applications to the asymptotic theory