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Application of homotopy perturbation method to nonlinear wave equations. (English) Zbl 1072.35502

Summary: The homotopy perturbation method is applied to the search for traveling wave solutions of nonlinear wave equations. Some examples are given to illustrate the determination of the periodic solutions or the bifurcation curves of the nonlinear wave equations.

MSC:

35A25 Other special methods applied to PDEs
35B10 Periodic solutions to PDEs
35B32 Bifurcations in context of PDEs
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[1] Deng, S.-F., Bäcklund transformation and soliton solutions for KP equation, Chaos, Solitons & Fractals, 25, 2, 475-480 (2005) · Zbl 1070.35059
[2] Tsigaridas, G.; Fragos, A.; Polyzos, I.; Fakis, M., Evolution of near-soliton initial conditions in non-linear wave equations through their Bäcklund transforms, Chaos, Solitons & Fractals, 23, 5, 1841-1854 (2005) · Zbl 1069.35066
[3] Pashaev, O.; Tanoğlu, G., Vector shock soliton and the Hirota bilinear method, Chaos, Solitons & Fractals, 26, 1, 95-105 (2005) · Zbl 1070.35056
[4] Vakhnenko, V. O.; Parkes, E. J.; Morrison, A. J., A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation, Chaos, Solitons & Fractals, 17, 4, 683-692 (2003) · Zbl 1030.37047
[5] De-Sheng, L.; Feng, G.; Hong-Qing, Z., Solving the (2+1)-dimensional higher order Broer-Kaup system via a transformation and tanh-function method, Chaos, Solitons & Fractals, 20, 5, 1021-1025 (2004) · Zbl 1049.35157
[6] Zayed, E. M.E.; Zedan, H. A.; Gepreel, K. A., Group analysis and modified extended tanh-function to find the invariant solutions and soliton solutions for nonlinear Euler equations, Int J Nonlinear Sci Numer Simul, 5, 3, 221-234 (2004) · Zbl 1069.35080
[7] Abdusalam, H. A., On an improved complex tanh-function method, Int J Nonlinear Sci Numer Simul, 6, 2, 99-106 (2005) · Zbl 1401.35012
[8] Abassy, T. A.; El-Tawil, M. A.; Saleh, H. K., The solution of KdV and mKdV equations using adomian pade approximation, Int J Nonlinear Sci Numer Simul, 5, 4, 327-340 (2004) · Zbl 1401.65122
[9] El-Sayed, S. M., The decomposition method for studying the Klein-Gordon equation, Chaos, Solitons & Fractals, 18, 5, 1025-1030 (2003) · Zbl 1068.35069
[10] Kaya, D.; El-Sayed, S. M., An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons & Fractals, 17, 5, 869-877 (2003) · Zbl 1030.35139
[11] Liu, H. M., Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method, Chaos, Solitons & Fractals, 23, 2, 573-576 (2005) · Zbl 1135.76597
[12] Liu, H. M., Variational approach to nonlinear electrochemical system, Int J Nonlinear Sci Numer Simul, 5, 1, 95-96 (2004)
[13] He, J. H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons & Fractals, 19, 4, 847-851 (2004) · Zbl 1135.35303
[14] Mesón, A. M.; Vericat, F., Variational analysis for the multifractal spectra of local entropies and Lyapunov exponents, Chaos, Solitons & Fractals, 19, 5, 1031-1038 (2004) · Zbl 1107.37021
[15] He, J. H., Variational iteration method: a kind of nonlinear analytical technique: some examples, Int J Nonlinear Mech, 34, 4, 699-708 (1999) · Zbl 1342.34005
[16] Drăgănescu, Gh.-E.; Căpălnăşan, V., Polycrystalline solids, nonlinear relaxation phenomena in polycrystalline solids, Int J Nonlinear Sci Numer Simul, 4, 3, 219-226 (2004)
[17] He, J. H., Modified Lindstedt-Poincare methods for some strongly nonlinear oscillations. Part I: expansion of a constant, Int J Nonlinear Mech, 37, 2, 309-314 (2002) · Zbl 1116.34320
[18] He, J. H., Modified Lindstedt-Poincare methods for some strongly nonlinear oscillations. Part II: a new transformation, Int J Nonlinear Mech, 37, 2, 315-320 (2002) · Zbl 1116.34321
[19] He, J. H., Modified Lindstedt-Poincare methods for some strongly nonlinear oscillations. Part III: double series expansion, Int J Nonlinear Sci Numer Simul, 2, 4, 317-320 (2001) · Zbl 1072.34507
[20] Liu, H. M., Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method, Chaos, Solitons & Fractals, 23, 577-579 (2005) · Zbl 1078.34509
[21] He, J. H., Asymptotic methods in engineering and sciences, or approximate analytical methods in engineering and sciences (2002), Henan Science and Technology Press: Henan Science and Technology Press Zhengzhou, Gongcheng Yu Kexue zhong de jinshi feixianxing feixi fangfa (in Chinese)
[22] He, J. H., Determination of limit cycles for strongly nonlinear oscillators, Phys Rev Lett, 90, 17, 174301 (2003)
[23] Shen, J.; Xu, W., Bifurcations of smooth and non-smooth travelling wave solutions of the Degasperis-Procesi equation, Int J Nonlinear Sci Numer Simul, 5, 4, 397-402 (2004) · Zbl 1401.35276
[24] Ma, S.; Lu, Q., Dynamical bifurcation for a predator-prey metapopulation model with delay, Int J Nonlinear Sci Numer Simul, 6, 1, 13-18 (2005) · Zbl 1401.92032
[25] Zhang, Y.; Xu, J., Classification and computation of non-resonant double hopf bifurcations and solutions in delayed van der Pol-Duffing system, Int J Nonlinear Sci Numer Simul, 6, 1, 63-68 (2005) · Zbl 1401.37058
[26] Zhang, Z.; Bi, Q., Bifurcations of a generalized Camassa-Holm equation, Int J Nonlinear Sci Numer Simul, 6, 1, 81-86 (2005) · Zbl 1401.37057
[27] Zheng, Y.; Fu, Y., Effect of damage on bifurcation and chaos of viscoelastic plates, Int J Nonlinear Sci Numer Simul, 6, 1, 87-91 (2005) · Zbl 1401.74196
[28] Nada, S. I., The effect of folding on covering spaces, immersions and bifurcation for chaotic manifolds, Int J Nonlinear Sci Numer Simul, 6, 2, 145-150 (2005) · Zbl 1401.37028
[29] He, J. H., Homotopy perturbation technique, Comp Meth Appl Mech Eng, 178, 257-262 (1999) · Zbl 0956.70017
[30] He, J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int J Nonlinear Mech, 35, 37-43 (2000) · Zbl 1068.74618
[31] He, J. H., Homotopy perturbation method for bifurcation of nonlinear problems, Int J Nonlinear Sci Numer Simul, 6, 2, 207-208 (2005) · Zbl 1401.65085
[32] Yao, Y., Abundant families of new traveling wave solutions for the coupled Drinfel’d-Sokolov-Wilson equation, Chaos, Solitons & Fractals, 24, 301-307 (2005) · Zbl 1062.35018
[33] El-Shahed, M., Application of He’s homotopy perturbation method to Volterra’s integro-differential equation, Int J Nonlinear Sci Numer Simul, 6, 2, 163-168 (2005) · Zbl 1401.65150
[34] He, J. H., Int J Nonlinear Sci Numer Simul, 4, 3, 313-314 (2003)
[35] Hao, T. H., Int J Nonlinear Sci Numer Simul, 4, 3, 311-312 (2003)
[36] He, J. H., Preliminary report on the energy balance for nonlinear oscillations, Mech Res Commun, 29, 2-3, 107-111 (2002) · Zbl 1048.70011
[37] Fu, Z.; Liu, S.; Liu, S., Multiple structures of 2-D nonlinear Rossby wave, Chaos, Solitons & Fractals, 24, 383-390 (2005) · Zbl 1067.35071
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