On billiard weak solutions of nonlinear PDE's and Toda flows
Abstract
A certain class of partial differential equations possesses singular solutions having discontinuous first derivatives ("peakons"). The time evolution of peaks of such solutions is governed by a finite dimensional completely integrable system. Explicit solutions of this system are constructed by using algebraic-geometric method which casts it as a flow on an appropriate Riemann surface and reduces it to a classical Jacobi inversion problem. The algebraic structure of the finite dimensional flow is also examined in the context of the Toda flow hierarchy. Generalized peakon systems are obtained for any simple Lie algebra and their complete integrability is demonstrated.
Cite
@article{arxiv.nlin/0005006,
title = {On billiard weak solutions of nonlinear PDE's and Toda flows},
author = {Mark Alber and Roberto Camassa and Michael Gekhtman},
journal= {arXiv preprint arXiv:nlin/0005006},
year = {2007}
}
Comments
12 pages, to be published in CRM Proc. & Lecture Notes, AMS, 25, 1--11, 2000