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The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE’s of shallow water and Dym type. (English) Zbl 1001.37062

This paper is devoted to the algebraic-geometric method for nonlinear integrable PDE’s and presents explicit theta-functional expressions for piecewise smooth weak solutions for a class of nonlinear PDE’s associated to nonlinear subvarieties of hyperelliptic Jacobians. The authors find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks are also obtained by using Jacobi inversion problems. The authors also relate their method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14H70 Relationships between algebraic curves and integrable systems
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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