SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Numerical dynamic detection, generation and simulation of solitary waves for nonlinear wave equations

N. Reguera and I. Alonso-Mallo

Abstract
There exist different techniques in the literature to obtain numerically solitary waves of nonlinear wave equations. In this work, we focus on the one known as iterative cleaning [3, 4]. It is based on numerical simulations of initial conditions that evolve with time into a single main pulse or train of pulses along with dispersive tails, and uses cleaning techniques in order to obtain a numerical approximation of a solitary wave. The main disadvantage of this method is that it is used in a specific way for each numerical experiment, determining in a manual way suitable times and regions for the cleaning of the numerical solution, which implies a lot of work. As an alternative, we propose a completely automatic algorithm that detects and generates solitary waves in an efficient and dynamic way [2]. It is based on a cleaning technique [1] and an efficient integration of the problem that checks symmetry conditions and the preservation of the velocity of the wave.

Bibliography
[1] I. Alonso-Mallo and N. Reguera, A self-adjusting algorithm for solitary wave simulations, to appear in Int. J. of Comput. Math.
[2] I. Alonso-Mallo and N. Reguera, Numerical detection and generation of solitary waves of nonlinear wave equations, submitted for publication.
[3] V. A. Dougalis, A. Durán, M. A. López-Marcos and D. E. Mitsotakis, A numerical study of the stability of solitary waves of the Bona-Smith family of Boussinesq systems, J. Nonlinear Sci., 17 (2007), pp. 569-607.
[4] H. Kalisch and J. L. Bona, Models for internal waves in deep water, Discret. Contin. Dyn. Sys., 6 no. 1 (2000), pp. 1-20.

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