Abstract
This paper is concerned with the numerical solution of the Cauchy problem for the Benjamin-Ono equationu t +uu x −Hu xx =0, whereH denotes the Hilbert transform. Our numerical method first approximates this Cauchy problem by an initial-value problem for a corresponding 2L-periodic problem in the spatial variable, withL large. This periodic problem is then solved using the Crank-Nicolson approximation in time and finite difference approximations in space, treating the nonlinear term in a standard conservative fashion, and the Hilbert transform by a quadrature formula which may be computed efficiently using the Fast Fourier Transform.
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References
L. Abdelouhab, J. L. Bona, M. Felland, and J.-C. Saut,Nonlocal models for nonlinear dispersive waves, Physica D, 40 (1989), pp. 360–392.
G. Akrivis, V. A. Dougalis, and O. Karakashian,Solving the systems of equations arising in the discretization of some nonlinear PDE's by implicit Runge-Kutta methods, RAIRO Modél. Math. Anal. Numér, 31 (1979), pp. 251–288.
T. B. Benjamin,Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), pp. 559–592.
P. L. Butzer and R. J. Nessel,Fourier Analysis and Approximation, I. One-dimensional Theory, Birkhäuser Verlag, Basel, 1971.
K. M. Case,Properties of the Benjamin-One equation, J. Math. Phys, 20 (1979), pp. 972–977.
R. J. Iório, Jr.,On the Cauchy problem for the Benjamin-Ono equation, Comm. P.D.E., 11 (1986), pp. 1031–1081.
R. L. James and J. A. C. Weideman,Pseudospectral methods for the Benjamin-Ono equation, in Advances in Computer Methods for Partial Differential Equations VII, R. Vichnevetsky, D. Knight, and G. Richter (eds.), IMACS, Brunswick N.J., 1992, pp. 371–377.
T. Miloh, M. Prestin, L. Shtilman, and M. P. Tulin,A note on the numerical and N-soliton solutions of the Benjamin-Ono evolution equation, Wave Motion, 17 (1993), pp. 1–10.
H. Ono,Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), pp. 1082–1091.
B. Pelloni and V. A. Dougalis,On spectral methods for the Benjamin-Ono equation, in HERMIS '96, Proceedings of the Third Hellenic-European Conference on Mathematics and Informatics, E. A. Lipitakis, ed., LEA, Athens, 1996, pp. 229–237.
R. D. Richtmyer and K. W. MortonDifference Methods for Initial-Value Problems, Wiley, New York, 1967.
N. J. Zabusky and M. D. Kruskal,Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), pp. 240–243.
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Communicated by Åke Björck.
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Thomée, V., Vasudeva Murthy, A.S. A numerical method for the Benjamin-Ono equation. Bit Numer Math 38, 597–611 (1998). https://doi.org/10.1007/BF02510262
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DOI: https://doi.org/10.1007/BF02510262