Numerical detection and generation of solitary waves for a nonlinear wave equation

• Automatic algorithm for detecting and generating solitary waves of nonlinear wave equations.
• Dynamic simulations of the solution of a nonlinear wave equation.
• Numerical approximation.
• Improvement of the iterative cleaning technique.
• Geometric integration of solitary waves.

This paper presents an automatic algorithm for detecting and generating solitary waves of nonlinear wave equations. With this purpose, dynamic simulations are carried out, the solution of which evolves into a main pulse along with smaller dispersive tails. The solitary waves are detected automatically by the algorithm by checking that they have constant amplitude and are symmetric respect to its maximum value. Once the main wave has been detected, the algorithm cleans the dispersive tails for time enough so that the solitary wave is obtained with the required precision.In order to use our algorithm, we need a spatial discretization with local basis. The numerical experiments are carried out for the BBM equation discretized in space with cubic finite elements along with periodic boundary conditions. Moreover, a geometric integrator in time is used in order to obtain good approximations of the solitary waves.