Soliton generation and nonlinear wave propagation

Abstract

Soliton generation by various means is described. First, experimental results of J . V. Wehausen and coworkers on solitons generated by a ship model in a towing tank are presented. Then T. Y. Wu’s related Boussinesq system of equations for shallow water motion with a moving pressure disturbance on the free surface is introduced. Numerical solutions of this system by D. M . W u and T. Y. Wu are shown to compare well with the experimental results. Similar numerical results on an initial-boundary value problem for the K .d.V . equation by C. K. Chu and coworkers are presented, which also yield soliton generation. Then J . P. Keener and J . Rinzel’s analysis of pulse generation in the Fitzhugh—Naqumo model of nerve conduction is described. Next, G. B. W hitham ’s modulation theory of nonlinear wave propagation is explained and the problem of relating its results to initial and boundary data is mentioned. Asym ptotic methods for solving this problem for the K .d.V . equation are described. They include the Lax—Levermore theory for the case of small dispersion, its extension by S. Venakides, and the centered simple wave solution of the modulation equations by A. V. G urevitch and L. P. Pitaevskii. Finally, the theory of weakly nonlinear waves of Choquet—Bruhat and of J . K. Hunter and the present author is described.