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A numerical study of compactons. (English) Zbl 0932.65096
Summary: The Korteweg-de Vries equation has been generalized by {\it P. Rosenau} and {\it J. M. Hyman} [Compactons: Solitons with finite wavelength, Phys. Rev. Lett. 70, No. 5, 564 (1993)] to a class of partial differential equations that has soliton solutions with compact support (compactons). Compactons are solitary waves with the remarkable soliton property that after colliding with other compactons, they re-emerge with the same coherent shape [loc. cit.]. In this paper, finite difference and finite element methods have been developed to study these types of equations. The analytical solutions and conserved quantities are used to assess the accuracy of these methods. A single compacton as well as the interaction of compactons have been studied. The numerical results have shown that these compactons exhibit true soliton behavior.

65M06Finite difference methods (IVP of PDE)
35Q51Soliton-like equations
35Q53KdV-like (Korteweg-de Vries) equations
35Q55NLS-like (nonlinear Schrödinger) equations
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
Full Text: DOI
[1] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Lecture Notes Series, London Mathematical Society, vol. 149, 1991 · Zbl 0762.35001
[2] Eilbeck, J. C.; Mcguire, G. R.: Numerical study of the regularized long-wave equation I: Numerical methods. J. comp. Phys. 19, 43-57 (1975) · Zbl 0325.65054
[3] A.R. Mitchell, D.F. Griffits, The Finite Difference Method in Partial Differential Equations, Wiley, New York, 1980
[4] R.D. Richtmyer, K.W. Morton, Difference Methods for Initial Value Problems, Wiley/Interscience, New York, 1967 · Zbl 0155.47502
[5] Rosenau, P.; Hyman, J. M.: Compactons: solitons with finite wavelengths. Phys. rev. Lett. 70, No. 5, 564 (1993) · Zbl 0952.35502
[6] Sanz-Serna, J. M.; Christie, I.: Petrov--Galerkin methods for nonlinear dispersive waves. J. comp. Phys. 39, 94-102 (1981) · Zbl 0451.65086
[7] Taha, T. R.: Numerical simulation of the KdV-mkdv equation. Int. J. Modern phys. 5, No. 2, 307 (1994) · Zbl 0940.65504
[8] Taha, T. R.; Ablowitz, M. J.: Analytical and numerical aspects of certain nonlinear evolution equations III, numerical Korteweg--de Vries equation. J. comp. Phys. 55, 231 (1984) · Zbl 0541.65083
[9] T.R. Taha, W. Schiesser, Method of lines solution of the K (2,2) (KdV-type) equation, in: A. Sydow (Ed.), Proceedings of the 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, vol. 2, Berlin, Germany, August 1997, pp. 127--130
[10] J.W. Thomas, Numerical partial differential equations, Text in Applied Mathematics, vol. 22, Springer, Berlin, 1995 · Zbl 0831.65087