×

zbMATH — the first resource for mathematics

Solving solitary waves with discontinuity by means of the homotopy analysis method. (English) Zbl 1071.76009
Summary: An analytic method, namely the homotopy analysis method, is applied to solve solitary waves governed by Camassa-Holm equation. Purely analytic solutions are given for soliton waves with and without continuity at crest. This provides with a new analytic approach to solve soliton waves with discontinuity.

MSC:
76B25 Solitary waves for incompressible inviscid fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
35Q51 Soliton equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Liao, S.J., Beyond perturbation: introduction to homotopy analysis method, (2003), Chapman Hall/CRC Boca Raton
[2] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl math comput, 147, 499-513, (2004) · Zbl 1086.35005
[3] Nayfeh, A.H., Perturbation methods, (2000), John Wiley & Sons New York · Zbl 0375.35005
[4] Lyapunov, A.M., General problem on stability of motion (English translation), (1992), Taylor & Francis London · Zbl 0786.70001
[5] Karmishin, A.V.; Zhukov, A.T.; Kolosov, V.G., Methods of dynamics calculation and testing for thin-walled structures (in Russian), (1990), Mashinostroyenie Moscow
[6] Adomian, G., Nonlinear stochastic differential equations, J math anal appl, 55, 441-452, (1975) · Zbl 0351.60053
[7] Liao, S.J.; Cheung, K.F., Homotopy analysis of nonlinear progressive waves in deep water, J eng math, 45, 2, 105-116, (2003) · Zbl 1112.76316
[8] Ayub, M.; Rasheed, A.; Hayat, T., Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int J eng sci, 41, 2091-2103, (2003) · Zbl 1211.76076
[9] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int J eng sci, 42, 123-135, (2004) · Zbl 1211.76009
[10] Wu, Y.; Wang, C.; Liao, S.J., Solving the one-loop soliton solution of the Vakhnenko equation by means of the homotopy analysis method, Chaos, solitons & fractals, 23, 5, 1733-1740, (2004) · Zbl 1069.35060
[11] Liao SJ. An analytical solution of unsteady boundary-layer flows caused by an impulsively stretching plate. Commun Nonlinear Sci Numer Simulat, in press · Zbl 1078.76022
[12] Camassa, R.; Holm, D.D., An integrable shallow water equation with peaked solitons, Phys rev lett, 71, 1661-1664, (1993) · Zbl 0972.35521
[13] Camassa, R.; Holm, D.D.; Hyman, J.M., A new integrable shallow water equation, (), 1-33 · Zbl 0808.76011
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.