Geometry of weak solutions of certain integrable nonlinear PDE's

摘要

We investigate the geometry of new classes of soliton-like weak solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm (1993) for their integrable shallow water equation. Alber, Camassa, Holm and Marsden (1994a) put this shallow water equation into the framework of complex integrable Hamiltonian systems on Riemann surfaces and use special limiting procedures to obtain new solutions such as quasiperiodic solutions, n-solitons, solitons with quasiperiodic background, billiard, and n-peakon solutions and complex angle representations for them. They also obtain explicit formulas for phase shifts of interacting soliton solutions using the method of asymptotic reduction of the corresponding angle representations. The method they use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow on N-dimensional quadrics. Amongst these geodesics, particularly interesting ones are the umbilic geodesics, which generate the class of umbilic soliton solutions. Umbilic solitons have the property that as the space variable x tends to infinity, the solution tends to a periodic wave, and as x tends to minus infinity, it tends to the same periodic wave with a phase shift. Elliptic billiards may be obtained from the problem of geodesics on quadrics by collapsing along the shortest semiaxis. The corresponding Hamiltonian billiard flows axe associated to new classes of solutions of equations in the Dym hierarchy. Such billiard type solutions have discontinuous spatial derivative and, thus, are weak solutions for this class of PDE's.