Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians

The algebraic–geometric approach is extended to study evolution equations associated with the energy-dependent Schrödinger operators having potentials with poles in the spectral parameter, in connection with Hamiltonian flows on nonlinear subvarieties of Jacobi varieties. The general approach is demonstrated by using new parametrizations for constructing quasi-periodic solutions of the shallow-water and Dym-type equations in terms of theta-functions. A qualitative description of real-valued solutions is provided.