×

Sinc numerical solution for solitons and solitary waves. (English) Zbl 1010.65043

Summary: A numerical scheme using the Sinc-Galerkin method is developed to approximate the solution for the Korteweg-de Vries model equation. Sinc approximation to both derivatives and indefinite integral reduce the integral equation to an explicit system of algebraic equations, then using various properties of Sinc functions, it is shown that the Sinc solution produce an error of order \(O(\exp(-c/h))\) for some positive constants \(c\), \(h\). The method is applied to a few test examples to illustrate the accuracy and the implementation of the method.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Al-Khaled, K., A sinc – galerkin approach on the p-system, Saitama math. J., 16, 1-13, (1998) · Zbl 1020.76040
[2] Courant, R.; Hilbert, D., Methods of mathematical physics, (1962), Wiley-Interscience New York · Zbl 0729.35001
[3] Drazin, P.G.; Johnson, R.S, Solitons: an introduction, (1989), Cambridge University Press Cambridge
[4] Ercolani, N.M.; Mclaughlin, D.W.; Roitner, H., Attractors and transients for a perturbed KdV equation, A nonlinear spectral analysis, J. nonlinear sci., 3, 477-539, (1993) · Zbl 0797.35145
[5] Bao-Feng, Feng; Taketomo, M., A finite difference method for the korteweg – de Vries and the kadomtsev – petviashvili equations, J. comput. appl. math., 1, 90, 97-118, (1998) · Zbl 0907.65085
[6] Gong, L.; Shen, S.S., Multiple supercritical solitary wave solution of the stationary forced korteweg – de Vries equation and their stability, SIAM J. appl. math., 54, 5, 1266-1290, (1994) · Zbl 0824.76012
[7] Grimshaw, R.; Mitsudera, H., Slowly varying solitary wave solutions of the perturbed korteweg – de Vries equation revisited, Stud. appl. math., 90, 75-86, (1993) · Zbl 0783.35062
[8] Kawahara, T.; Toh, S., Nonlinear dispersive periodic waves in the presence of instability and damping, Phys. fluids, 28, 1636-1638, (1985)
[9] Kodama, Y.; Ablowitz, M., Perturbations of solitons and solitary waves, Stud. appl. math., 64, 225-245, (1994) · Zbl 0486.76029
[10] Logan, D.J., An introduction to nonlinear partial differential equations, (1994), Wiley New York · Zbl 0834.35001
[11] Ogawa, T.; Suzuki, H., On the spectra of pulses in a nearly integrable system, SIAM J. appl. math., 57, 2, 485-500, (1997) · Zbl 0873.35082
[12] Ogawa, T., Traveling wave solutions to a perturbed korteweg – de Vries equation, Hiroshima math. J., 24, 401-422, (1994) · Zbl 0812.76015
[13] Stenger, F., Numerical methods based on sinc and analytic functions, (1993), Springer New York · Zbl 0803.65141
[14] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley-Interscience New York · Zbl 0373.76001
[15] Winther, R., A conservative finite element method for the korteweg – de Vries equation, Math. comput., 34, 23-43, (1980) · Zbl 0422.65063
[16] Vanden-Broeck, J.M., Free-surface flow over a semi-circular obstruction in a channel, Phys. fluids, 30, 2315-2317, (1987) · Zbl 0635.76013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.