We propose a simple, though powerful, technique for numerical solutions of the Benjamin-Ono equation. This approach is based on a global collocation method using Sinc basis functions. Some properties of the Sinc collocation method required for our subsequent development are given and utilized to reduce the computation of the Benjamin-Ono equation to a system of ordinary differential equations. The propagation of one soliton and the interaction of two solitons are used to validate our numerical method. The method is easy to implement and yields accurate results. 1. Introduction It is well known that nonlinear partial differential equations (NPDEs) are widely used to describe complex phenomena in various fields of sciences, such as physics, biology, and chemistry. In this paper, we consider the initial value problem for the Benjamin-Ono (BO) equation of the form together with the following initial and the boundary conditions: Here is a real valued function and denotes the Hilbert transform defined and described by [1] (amongst others): where denotes the Cauchy principal value. This equation was derived by Benjamin [2] and later by Ono [3] as a model for one-dimensional waves in deep water and it has a close relation with the famous KdV equation which models long waves in shallow water [4]. We recall that the BO equation has an infinite sequence of invariants [5], the first three of which are The BO has been shown to admit rational analytical soliton solutions in and [6]. However, for a given arbitrary initial condition, finding analytical solutions of the BO equation becomes an intractable problem. Therefore the use of numerical methods plays an important role in the study of the dynamics of the BO equation. James and Weideman [7] used the Fourier method which implicitly assumes the periodicity of the boundary conditions and a method based on rational approximating function to compute numerical solutions of the BO. The rational method was shown to have spectacular accuracy for solutions that do not wander too far from the origin while, for long dated solutions, the Fourier method was shown to retain superior accuracy. Miloh et al. [8] proposed an efficient pseudospectral method for the numerical solution of the weakly nonlinear Benjamin-Ono equation for arbitrary initial conditions and suggested a practical new relationship for estimating the number of solitons in terms of arbitrary initial conditions. Thomée and Vsaudeva Murthy [9] used the Crank-Nicolson approximation in time and finite difference approximations in space to solve the BO equation. They
References
[1]
J. A. C. Weideman, “Computing the Hilbert transform on the real line,” Mathematics of Computation, vol. 64, no. 210, pp. 745–762, 1995.
[2]
T. B. Benjamin, “Internal waves of permanent form in fluids of great depth,” Journal of Fluid Mechanics, vol. 29, no. 3, pp. 559–592, 1967.
[3]
H. Ono, “Algebraic solitary waves in stratified fluids,” Journal of the Physical Society of Japan, vol. 39, no. 4, pp. 1082–1091, 1975.
[4]
C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,” Communications on Pure and Applied Mathematics, vol. 46, no. 4, pp. 527–620, 1993.
[5]
L. Abdelouhab, J. L. Bona, M. Felland, and J. Saut, “Nonlocal models for nonlinear, dispersive waves,” Physica D, vol. 40, no. 3, pp. 360–392, 1989.
[6]
J. Satsuma and Y. Ishimori, “Periodic wave and rational soliton solutions of the Benjamin-Ono equation,” Journal of the Physical Society of Japan, vol. 46, no. 2, pp. 681–687, 1979.
[7]
R. L. James and J. A. C. Weideman, “Pseudospectral methods for the Bejamin-Ono equation,” in Advances in Computer Methods for Partial Differential Equations VII, pp. 371–377, 1992.
[8]
T. Miloh, M. Prestin, L. Shtilman, and M. P. Tulin, “A note on the numerical and -soliton solutions of the Benjamin-Ono evolution equation,” Wave Motion, vol. 17, no. 1, pp. 1–10, 1993.
[9]
V. Thomée and A. S. Vasudeva Murthy, “A numerical method for the Benjamin-Ono equation,” BIT Numerical Mathematics, vol. 38, no. 3, pp. 597–611, 1998.
[10]
J. P. Boyd and Z. Xu, “Comparison of three spectral methods for the Benjamin-Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions,” Wave Motion, vol. 48, no. 8, pp. 702–706, 2011.
[11]
F. Stenger, “A Sinc-Galerkin methods for solution of boundary value problems,” Mathematics of Computation, vol. 33, pp. 85–109, 1979.
[12]
K. Al-Khaled, “Sinc numerical solution for solitons and solitary waves,” Journal of Computational and Applied Mathematics, vol. 130, no. 1-2, pp. 283–292, 2001.
[13]
R. Mokhtari and M. Mohammadi, “Numerical solution of GRLW equation using Sinc-collocation method,” Computer Physics Communications, vol. 181, no. 7, pp. 1266–1274, 2010.
[14]
J. Lund and K. L. Bowers, Sinc Methods For Quadrature and Differential Equations, SIAM, Philadelphia, Pa, USA, 1992.
[15]
F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, NY, USA, 1993.
[16]
Y. Matsuno, “Exact multi-soliton solution of the Benjamin-Ono equation,” Journal of Physics A: Mathematical and General, vol. 12, no. 4, pp. 619–621, 1979.
[17]
Y. Matsuno, “Interaction of the Benjamin-Ono solitons,” Journal of Physics A: Mathematical and General, vol. 13, no. 5, pp. 1519–1536, 1980.
