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Shock-peakon and shock-compacton solutions for \(K(p,q)\) equation by variational iteration method. (English) Zbl 1119.65099
Summary: By the variational iteration method, we obtain new solitary solutions for nonlinear dispersive equations. Particularly, shock-peakon solutions in the \(K(2,2)\) equation and shock-compacton solutions in the \(K(3,3)\) equation are found by this simple method. These two types of solutions are new solitary wave solutions which have the shapes of shock solutions and compacton solutions (or peakon solutions).

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
35L67 Shocks and singularities for hyperbolic equations
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[1] Camassa, R.; Holm, D., Phys. rev. lett., 71, 1661-1664, (1993)
[2] Ding, D.P.; Tian, L.X., The study of solution of dissipative camassa – holm equation on total space, Internat. J. nonlinear sci., 1, 1, 37-42, (2006) · Zbl 1394.35409
[3] He, J.H., Variational iteration method—a kind of non-linear analytical technique: some examples, Internat. J. non-linear mech., 34, 4, 699-708, (1999) · Zbl 1342.34005
[4] He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl. math. comput., 114, 2-3, 115-123, (2000) · Zbl 1027.34009
[5] He, J.H., Some asymptotic methods for strongly nonlinear equations, Internat. J. modern phys. B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039
[6] J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation, de-Verlag im Internet GmbH, Berlin, 2006.
[7] He, J.H.; Wu, X.H., Construction of solitary solution and compacton solution by variational iteration method, Chaos, solitons fractals, 29, 108-113, (2006) · Zbl 1147.35338
[8] Ju, L., On solution of the dullin – gottwald – holm equation, Internat. J. nonlinear sci., 1, 1, 43-48, (2006) · Zbl 1394.35419
[9] Rosenau, P.; Hyman, J.M., Compactons: solitons with finite wavelengths, Phys. rev. lett., 75, 564-567, (1993) · Zbl 0952.35502
[10] Tian, L.X.; Fang, G.C.; Gui, G.L., Well-posedness and blow-up for an integrable shallow water equation with strong dispersive term, Internat. J. nonlinear sci., 1, 1, 3-13, (2006) · Zbl 1394.35439
[11] Tian, L.X.; Yin, J.L., New compacton solutions and solitary wave solutions of fully nonlinear generalized camassa – holm equations, Chaos, solitons fractals, 20, 2, 289-299, (2004) · Zbl 1046.35101
[12] Tian, L.X.; Yin, J.L., Stability of multi-compacton solutions and backlund transformation in K(m,n,l), Chaos, solitons fractals, 23, 1, 159-169, (2005) · Zbl 1075.37028
[13] Tian, L.X.; Yin, J.L., New compacton and peakon solitary wave solutions of fully nonlinear sine-Gordon equation, Chaos, solitons fractals, 24, 1, 353-363, (2005) · Zbl 1067.35093
[14] Wazwaz, A.M., New compactons, solitons and periodic solutions for nonlinear variants of the KdV and the KP equations, Chaos, solitons fractal, 22, 1, 249-260, (2004) · Zbl 1062.35121
[15] Wazwaz, A.M., Existence and construction of compacton solutions, Chaos, solitons fractals, 19, 3, 463-470, (2004) · Zbl 1068.35124
[16] Wazwaz, A.M., Compacton, soliton and periodic solutions for some forms of nonlinear klein – gordon equations, Chaos, solitons fractals, 28, 1005-1013, (2006) · Zbl 1099.35125
[17] Wazwaz, A.M.; Helal, M.A., Variants of the generalized fifth-order KdV equation with compact and noncompact structures, Chaos, solitons fractals, 21, 3, 579-589, (2004) · Zbl 1049.35163
[18] Wazwaz, A.M.; Helal, M.A., Nonlinear variants of the BBM equation with compact and noncompact physical structures, Chaos, solitons fractals, 26, 767-776, (2005) · Zbl 1078.35110
[19] Whitbam, G.B., Linear and nonlinear waves, (1974), Wiley New York
[20] Zhu, Y.; Chang, Q.; Wu, S., Exact solitary-wave solutions with compact support for the modified KdV equation, Chaos, solitons fractals, 24, 1, 365-369, (2005) · Zbl 1067.35099
[21] Zhu, Y.; Gao, X., Exact special solitary solutions with compact support for the nonlinear dispersive K(m,n) equations, Chaos, solitons fractals, 27, 2, 487-493, (2006) · Zbl 1088.35547
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