New completely integrable evolution equations
Roberto Camassa, LOs Alamos
A new class of completely integrable nonlinear PDE's has recently been
derived in the context of shallow water wave motion and front
propagation in nematic crystals. The class of solutions of these
equations is surprisingly rich, and it includes solutions with loss of
smoothness in finite times and weak soliton solutions. By casting these
equations in the framework of complex Hamiltonian systems on Riemann
surfaces, and using special limiting procedures, one obtains a unifying
geometric point of view. In particular, through this approach, an
unexpected connection is found between solutions of the PDE's and the
classical problem of geodesic flow on the surface of n-dimensional
quadrics. Weak solutions of PDE's, which admit discontinuities in the
first derivatives, can be associated with the limiting case in which
geodesics on quadrics degenerate into elliptic and hyperbolic
billiards. Integrability (by quadrature) of the system of ode's
governing the position and amplitude of the jumps follows naturally
form this approach.
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