Numerical Study of the Second-Order Correct Hamiltonian Model for Unidirectional Water Waves

Abstract

Second-order correct versions of the usual KdV–BBM models for unidirectional propagation of long-crested, surface water waves are considered here. The class of models studied here has a Hamiltonian structure and, in certain circumstances, is globally well posed. A fully discrete, numerical algorithm based on the Fourier-spectral method is developed and its convergence tested. We then use this algorithm to generate solitary-wave solutions to the model. While such waves are known to exist, exact formulas for them are not available. The heart of the paper is a sequence of numerical experiments aimed at understanding the stability of individual solitary waves, their interaction, and whether or not the model exhibits resolution of general initial data into solitary waves. A comparison is made between the first-order correct KdV–BBM models and the associated second-order correct equations. A number of tentative conjectures pertaining to the models are put forward on the basis of these experiments.

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Acknowledgments

JB, HC, and YH are all grateful for support and excellent working conditions from the Mathematics Department of the University of Tennessee at Knoxville. Part of this work was done, while JB and HC were visiting professors at Victoria University inWellington, New Zealand. YH thanks the Simons Foundation for travel support during the period of this collaboration. OK was partially supported by NSF grant DMS-1620288.

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Correspondence to Jerry L. Bona.

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Bona, J.L., Chen, H., Hong, Y. et al. Numerical Study of the Second-Order Correct Hamiltonian Model for Unidirectional Water Waves. Water Waves 1, 3–40 (2019). https://doi.org/10.1007/s42286-019-00003-y

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Keywords

  • Higher-order water wave models
  • Hamiltonian models
  • Solitary Waves
  • Spectral methods
  • KdV-BBM models