On billiard weak solutions of nonlinear PDE's and Toda flows

Mark Alber; Roberto Camassa; Michael Gekhtman

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arXiv · nlin/0005006

On billiard weak solutions of nonlinear PDE's and Toda flows

Mark Alber·Roberto Camassa·Michael Gekhtman

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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.

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Mark Alber, Roberto Camassa, Michael Gekhtman. 2000-05-02. On billiard weak solutions of nonlinear PDE's and Toda flows. https://arxiv.org/abs/nlin/0005006

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