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Numerical approximation of solitary waves of the Benjamin equation

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  • Dougalis, V.A.
  • Durán, A.
  • Mitsotakis, D.E.

Abstract

This paper presents several numerical techniques to generate solitary-wave profiles of the Benjamin equation. The formulation and implementation of the methods focus on some specific points of the problem: on the one hand, the approximation of the nonlocal term is accomplished by Fourier techniques, which determine the spatial discretization used in the experiments. On the other hand, in the numerical continuation procedure suggested by the derivation of the model and already discussed in the literature, several algorithms for solving the nonlinear systems are described and implemented: the Petviashvili method, the Preconditioned Conjugate Gradient Newton method and two Squared-Operator methods. A comparative study of these algorithms is made in the case of the Benjamin equation; Newton's method combined with Preconditioned Conjugate Gradient techniques, emerges as the most efficient. The resulting numerical profiles are shown to have a high accuracy as travelling-wave solutions when they are used as initial conditions in a time-stepping procedure for the Benjamin equation. The paper also explores the generation of multi-pulse solitary waves.

Suggested Citation

  • Dougalis, V.A. & Durán, A. & Mitsotakis, D.E., 2016. "Numerical approximation of solitary waves of the Benjamin equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 127(C), pages 56-79.
    • Handle: RePEc:eee:matcom:v:127:y:2016:i:c:p:56-79
    DOI: 10.1016/j.matcom.2012.07.008
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    1. Boyd, John P., 2007. "Why Newton’s method is hard for travelling waves: Small denominators, KAM theory, Arnold’s linear Fourier problem, non-uniqueness, constraints and erratic failure," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(2), pages 72-81.
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