On asymptotically equivalent shallow water wave equations

Details On asymptotically equivalent shallow water wave equations On asymptotically equivalent shallow water wave equations

2003-07-04

arxiv.org/abs/nlin/0307011v1

Physica D, 190:1--14, 2004

Nonlinear Sciences

Pattern Formation and Solitons

Scientific paper

10.1016/j.physd.2003.11.004

The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it family} of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama-Helmholtz transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de Vries equation, and a generalization of the Camassa-Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to 5th order. The travelling wave solutions of the CH equation are shown to agree to 5th order with the exact solution.

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