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February  2003, 3(1): 115-139. doi: 10.3934/dcdsb.2003.3.115

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

Roberto Camassa 1,

1. 

Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States

Received  March 2002 Revised  September 2002 Published  November 2002

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: lattice dynamics., Hamiltonian structures, Completely integrable systems.
Mathematics Subject Classification: 37K1.
Citation: Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115
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