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Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method. (English) Zbl 1139.76013
Summary: We use the homotopy analysis method to find a family of solitary smooth hump solutions of Camassa-Holm equation. This approximate solution, which is obtained as a series in exponentials, agrees well with the known exact solution. This paper complements the work of W. Wu and S. Liao [ibid. 26, No. 1, 177–185 (2005; Zbl 1071.76009)] who used the homotopy analysis method to find a different family of solitary-wave solutions.

##### MSC:
 76B25 Solitary waves for incompressible inviscid fluids 76M25 Other numerical methods (fluid mechanics) (MSC2010) 35Q51 Soliton equations
series solution
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##### References:
 [1] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton [2] Liao, S.J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transfer, 48, 2529-2539, (2005) · Zbl 1189.76142 [3] Liao, S.J., Series solutions of unsteady boundary-layer flows over a stretching flat plate, Stud appl math, 117, 239-264, (2006) · Zbl 1145.76352 [4] Liao, S.J.; Magyari, E., Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Z angew math phys (ZAMP), 57, 777-792, (2006) · Zbl 1101.76056 [5] Liao, S.J.; Su, J.; Chwang, A.T., Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body, Int J heat mass transfer, 49, 2437-2445, (2006) · Zbl 1189.76549 [6] Tan, Y.; Xu, H.; Liao, S.J., Explicit series solution of travelling waves with a front of Fisher equation, Chaos, solitons & fractals, 31, 462-472, (2007) · Zbl 1143.35313 [7] Wu, W.; Liao, S.J., Solving solitary waves with discontinuity by means of the homotopy analysis method, Chaos, solitons & fractals, 26, 177-185, (2005) · Zbl 1071.76009 [8] Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys lett A, 360, 109-113, (2006) · Zbl 1236.80010 [9] Abbasbandy, S., The application of homotopy analysis method to solve a generalized hirota – satsuma coupled KdV equation, Phys lett A, 361, 478-483, (2007) · Zbl 1273.65156 [10] Abbasbandy, S., Homotopy analysis method for heat radiation equations, Int commun heat mass transfer, 34, 380-387, (2007) [11] Abbasbandy S, Samadian Zakaria F. Soliton solutions for the fifth-order KdV equation with the homotopy analysis method. Nonlinear Dyn, 2008, doi:10.1007/s11071-006-9193-y. · Zbl 1170.76317 [12] Hayat, T.; Khan, M., Homotopy solutions for a generalized second-grade fluid past a porous plate, Nonlinear dyn, 42, 395-405, (2005) · Zbl 1094.76005 [13] Hayat, T.; Khan, M.; Ayub, M., On non-linear flows with slip boundary condition, Z angew math phys (ZAMP), 56, 1012-1029, (2005) · Zbl 1097.76007 [14] Liao, S.J.; Cheung, K., Homotopy analysis of nonlinear progressive waves in deep water, J eng math, 45, 105-116, (2003) · Zbl 1112.76316 [15] Sajid, M.; Hayat, T.; Asghar, S., On the analytic solution of the steady flow of a fourth grade fluid, Phys lett A, 355, 18-26, (2006) [16] Tan, Y.; Abbasbandy, S., Homotopy analysis method for quadratic Riccati differential equation, Commun nonlinear sci numer simul, 13, 539-546, (2008) · Zbl 1132.34305 [17] Wang, C., Analytic solutions for a liquid film on an unsteady stretching surface, Heat mass transfer, 42, 759-766, (2006) [18] He, J.H., Comparison of homotopy perturbation method and homotopy analysis method, Appl math comput, 156, 527-539, (2004) · Zbl 1062.65074 [19] Liao, S.J., Comparison between the homotopy analysis method and homotopy perturbation method, Appl math comput, 169, 1186-1194, (2005) · Zbl 1082.65534 [20] Hayat, T.; Sajid, M., On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Phys lett A, 361, 316-322, (2007) · Zbl 1170.76307 [21] Camassa, R.; Holm, D.D., An integrable shallow water equation with peaked solitons, Phys rev lett, 71, 1661-1664, (1993) · Zbl 0972.35521 [22] Dai, H.H., Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods, Wave motion, 28, 367-381, (1998) · Zbl 1074.74541 [23] Boyd, J.P., Near-corner waves of the camassa – holm equation, Phys lett A, 336, 342-348, (2005) · Zbl 1136.35445 [24] Camassa, R.; Holm, D.D.; Hyman, J.M., A new integrable shallow water equation, Adv appl mech, 31, 1-33, (1994) · Zbl 0808.76011 [25] Camassa, R., Characteristics and the initial value problem of a completely integrable shallow water equation, Discrete contin dyn syst ser B, 3, 115-139, (2003) · Zbl 1031.37063 [26] Chen, C.; Tang, M., A new type of bounded waves for degasperis – procesi equation, Chaos, solitons & fractals, 27, 698-704, (2006) · Zbl 1082.35044 [27] Lenells, J., Traveling wave solutions of the camassa – holm equation, J differen equat, 217, 393-430, (2005) · Zbl 1082.35127 [28] Mustafa, O.G., A note on the degasperis – procesi equation, J nonlinear math phys, 12, 10-14, (2005) · Zbl 1067.35078 [29] Parker, A., On the camassa – holm equation and a direct method of solution: I. bilinear form and solitary waves, Proc roy soc London A, 460, 2929-2957, (2004) · Zbl 1068.35110 [30] Parkes, E.J.; Vakhnenko, V.O., Explicit solutions of the camassa – holm equation, Chaos, solitons & fractals, 26, 1309-1316, (2005) · Zbl 1072.35156 [31] Qian, T.; Tang, M., Peakons and periodic cusp waves in a generalized camassa – holm equation, Chaos, solitons & fractals, 12, 1347-1360, (2001) · Zbl 1021.35086 [32] Shen, J.; Xu, W.; Li, W., Bifurcations of travelling wave solutions in a new integrable equation with peakon and compactons, Chaos, solitons & fractals, 27, 413-425, (2006) · Zbl 1094.35109 [33] Tian, L.; Song, X., New peaked solitary wave solutions of the generalized camassa – holm equation, Chaos, solitons & fractals, 19, 621-637, (2004) · Zbl 1068.35123 [34] Wazwaz, A., New solitary wave solutions to the modified forms of degasperis – procesi and camassa – holm equations, Appl math comput, 186, 130-141, (2007) · Zbl 1114.65124
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