Solitary waves of the EW and RLW equations. (English) Zbl 1152.35478

Ramos, J. I.

Chaos Solitons Fractals 34, No. 5, 1498-1518 (2007).

Summary: Eight finite difference methods are employed to study the solitary waves of the equal-width (EW) and regularized long-wave (RLW) equations. The methods include second-order accurate (in space) implicit and linearly implicit techniques, a three-point, fourth-order accurate, compact operator algorithm, an exponential method based on the local integration of linear, second-order ordinary differential equations, and first- and second-order accurate temporal discretizations. It is shown that the compact operator method with a Crank-Nicolson discretization is more accurate than the other seven techniques as assessed for the three invariants of the EW and RLW equations and the \(L_{2}\)-norm errors when the exact solution is available. It is also shown that the use of Gaussian initial conditions may result in the formation of either positive or negative secondary solitary waves for the EW equation and the formation of positive solitary waves with or without oscillating tails for the RLW equation depending on the amplitude and width of the Gaussian initial conditions. In either case, it is shown that the creation of the secondary wave may be preceded by a steepening and an narrowing of the initial condition. The creation of a secondary wave is reported to also occur in the dissipative RLW equation, whereas the effects of dissipation in the EW equation are characterized by a decrease in amplitude, an increase of the width and a curving of the trajectory of the solitary wave. The collision and divergence of solitary waves of the EW and RLW equations are also considered in terms of the wave amplitude and the invariants of these equations.

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35Q51	Soliton equations

References:

[1]	Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Philos Trans Roy Soc (London), Series A, 272, 47-78 (1972) · Zbl 0229.35013
[2]	Peregrine, D. H., Calculations of the development of an undular bore, J Fluid Mech, 25, 321-330 (1966)
[3]	Peregrine, D. H., Long waves on a beach, J Fluid Mech, 27, 815-827 (1967) · Zbl 0163.21105
[4]	Dodd, R. K.; Eilbeck, J. C.; Gibbon, J. D.; Morris, H. C., Solitons and nonlinear wave equations (1982), Academic Press: Academic Press New York · Zbl 0496.35001
[5]	Lewis, J. C.; Tjon, J. A., Resonant production of solitons in the RLW equation, Phys Lett A, 73, 275-279 (1979)
[6]	Raslan, K. R., Collocation method using quartic B-spline for the equal width (EW) equation, Appl Math Comput, 168, 795-805 (2005) · Zbl 1082.65583
[7]	Gardner, L. R.T.; Gardner, G. A.; Dogan, A., A least squares finite element scheme for the RLW equation, Commun Numer Meth Eng, 12, 795-804 (1996) · Zbl 0867.76040
[8]	Sloan, D. M., Fourier pseudospectral solution of the regularised long wave equation, J Comput Appl Math, 36, 159-179 (1991) · Zbl 0732.65096
[9]	Araújo, A.; Durán, A., Error propagation in the numerical integration of solitary waves. The regularized long wave equation, Appl Numer Math, 36, 197-217 (2001) · Zbl 0972.65062
[10]	Durán, A.; López-Marcos, M. A., Conservative numerical methods for solitary wave interactions, J Phys A, 36, 7761-7770 (2003) · Zbl 1038.35091
[11]	Bona, J. L.; Soyeur, A., On the stability of solitary wave solutions of model equations for long waves, J Nonlinear Sci, 4, 449-470 (1994) · Zbl 0809.35095
[12]	Gardner, L. R.T.; Gardner, G. A., Solitary waves of the regularised long-wave equation, J Comput Phys, 91, 441-459 (1990) · Zbl 0717.65072
[13]	Luo, Z.; Liu, R., Mixed finite element analysis and numerical solution for the RLW equation, SIAM J Numer Anal, 36, 89-104 (1998) · Zbl 0927.65123
[14]	Jain, P. C.; Shankar, R.; Singh, T. V., Numerical solution of the regularized long-wave equation, Commun Numer Meth Eng, 9, 579-586 (1993) · Zbl 0779.65062
[15]	Zaki, S. I., Solitary waves of the splitted RLW equation, Comput Phys Commun, 138, 80-91 (2001) · Zbl 0984.65103
[16]	Dağ, I.; Saka, B.; Irk, D., Galerkin method for the numerical solution of the RLW equation using quintic B-splines, J Comput Appl Math, 190, 532-547 (2006) · Zbl 1086.65094
[17]	El-Danaf, T. S.; Ramadan, M. A.; Abd Alaal, F. E.I., The use of Adomian decomposition method for solving the regularized long-wave equation, Chaos, Solitons & Fractals, 26, 747-757 (2005) · Zbl 1073.35010
[18]	Kaya, D.; El-Sayed, S. M., An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons & Fractals, 17, 869-877 (2003) · Zbl 1030.35139
[19]	Shivamoggi, B. K.; Rollins, D. K., Evolution of solitary-wave solution of the perturbed regularized long-wave equation, Chaos, Solitons & Fractals, 13, 1129-1136 (2002) · Zbl 1031.76012
[20]	Ramos, J. I., Implicit, compact linearized, \(θ\)-methods with factorization for multidimensional reaction-diffusion equations, Appl Math Comput, 94, 17-43 (1998) · Zbl 0943.65098
[21]	Ramos, J. I., On diffusive methods and exponentially-fitted techniques, Appl Math Comput, 103, 69-96 (1999) · Zbl 0929.65057
[22]	Ramos, J. I., A smooth locally-analytical technique for singularly perturbed two-point boundary value problems, Appl Math Comput, 163, 1123-1142 (2005) · Zbl 1067.65074
[23]	Ramos, J. I., Linearization methods in classical and quantum mechanics, Comput Phys Commun, 153, 199-208 (2003) · Zbl 1196.81114
[24]	Bryan, A. C.; Stuart, A. E.G., Solitons and the regularized long wave equation: a nonexistence theorem, Chaos, Solitons & Fractals, 7, 1881-1886 (1996) · Zbl 1080.35528