Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Characteristics and the initial value problem of a completely integrable shallow water equation

Pages: 115 - 139, Volume 3, Issue 1, February 2003      doi:10.3934/dcdsb.2003.3.115

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Roberto Camassa - Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States (email)

Abstract: The initial value problem for a completely integrable shallow water wave equation is analyzed through its formulation in terms of characteristics. The resulting integro-differential equations give rise to finite dimensional projections onto a class of distributional solutions of the equation, equivalent to taking the Riemann sum approximation of the integrals. These finite dimensional projections are then explicitly solved via a Gram-Schmidt orthogonalization procedure. A particle method based on these reductions is implemented to solve the wave equation numerically.

Keywords:  Completely integrable systems, Hamiltonian structures, lattice dynamics.
Mathematics Subject Classification:  37K15.

Received: March 2002;      Revised: September 2002;      Available Online: November 2002.