Article

Letters in Mathematical Physics

, Volume 32, Issue 2, pp 137-151

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

  • Mark S. AlberAffiliated withSchool of Mathematics, Institute for Advanced Study, Princeton and Department of Mathematics, University of Notre Dame
  • , Roberto CamassaAffiliated withCenter for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory
  • , Darryl D. HolmAffiliated withCenter for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory
  • , Jerrold E. MarsdenAffiliated withDepartment of Mathematics, University of California

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Abstract

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.

Mathematics Subject Classifications (1991)

58F07 70H99 76B15