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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
##### Keywords:
Camassa-Holm equation
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##### 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
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