Alber, Mark S. and Camassa, Roberto and Holm, Darryl D. and Marsden, Jerrold E.
(1995)
*On the Link between Umbilic Geodesics and Soliton Solutions of Nonlinear PDEs.*
Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences, 450
(1940).
pp. 677-692.
ISSN 1364-5021.
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## Abstract

In this paper we describe a new class of soliton solutions, called umbilic solitons, for certain nonlinear integrable PDES. These 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 a phase shifted wave of the same shape. The equations admitting solutions in this new class include the Dym equation and equations in its hierarchy. The methods used to find and analyse these solutions are those of algebraic and complex geometry. We look for classes of solutions by constructing associated finite-dimensional integrable Hamiltonian systems on Riemann surfaces. In particular, in this setting we use geodesics on n-dimensional quadrics to find the spatial, or x-flow, which, together with the commuting t-flow given by the equation itself, defines new classes of solutions. Amongst these geodesics, particularly interesting ones are the umbilic geodesics, which then generate the class of umbilic soliton solutions. This same setting also enables us to introduce another class of solutions of Dym-like equations, which are related to elliptic and umbilic billiards.

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Additional Information: | © 1995 The Royal Society. Received 30 November 1994; accepted 26 April 1995. We thank Nick Ercolani and Carlos Tomei for useful discussions on elliptic billiards. Mark Alber also thanks the Institute for Advanced Study in Princeton and the Center for Nonlinear Studies at Los Alamos National Laboratory for their hospitality during the Fall of 1993 and during three visits in August 1993 and January and June 1994. Research by M.S.A. was partly supported by NSF grants DMS 9403861 and 9022140, research by R.C. and D.D.H. was partly supported by the DOE, CHAMMP and HPCC programmes and research by J.E.M. was partly supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences. | ||||||||||||||||||

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