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Exact explicit peakon and periodic cusp wave solutions for several nonlinear wave equations. (English) Zbl 1178.34002
The author considers the existence of travelling wave solutions to the following six partial differential equations: Vakhnenko, Ostrovsky, Camasso-Holm (generalized), Rosenau-Hyman, KdV (“more realistic”), perturbed Boussinesq. The usual ansatz \(u(x,t)=\varphi(x+ct)\) leads to boundary value problems for ordinary differential equations. All these problems can be reduced to the study of the phase portrait of planar autonomous systems. By exploiting special properties of these systems, the author is able to present exact explicit parametric representations of the traveling wave solutions.

34A05 Explicit solutions, first integrals of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
35L05 Wave equation
35Q51 Soliton equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
Full Text: DOI
[1] Byrd P.F., Fridman M.D.: Handbook of Elliptic Integrals for Engineers and Sciensists. Springer, Berlin (1971)
[2] Camassa R., Holm D.D., Hyman J.M.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994) · Zbl 0808.76011 · doi:10.1016/S0065-2156(08)70254-0
[3] Jibin L., Huihui D.: On the Study of Singular Nonlinear Travelling Wave Equations: Dynamical System Approach. Science Press, Beijing (2007)
[4] Jibin L., Jiahong W., Huaiping Z.: Travelling waves for an integrable higher order KdV type wave Equations. Int. J. Bifurcat. Chaos 16(8), 2235–2260 (2006) · Zbl 1192.37100 · doi:10.1142/S0218127406016033
[5] Jibin, L., Guanrong, C.: On a class of singular nonlinear travelling wave equations. Int. J. Bifurcat. Chaos 17(11), 4049–4065 (2007) · Zbl 1158.35080 · doi:10.1142/S0218127407019858
[6] Li Y.A.: Weak solutions of generalized Boussinesq system. J. Dyn. Diff. Equat. 11(4), 625–669 (1999) · Zbl 0941.35078 · doi:10.1023/A:1022611428785
[7] Lopes O.: Stability of peakons for the generalized Camassa-Holm equation. Electro. J. Diff. Equat. 2002(5), 1–12 (2002) · Zbl 1088.35060
[8] Morrison T.P., Parkes E.J., Vakhnenko V.O.: The N-loop soliton solution of the Vakhnenko equation. Nonlinearity 12, 1427–1437 (1999) · Zbl 0935.35129 · doi:10.1088/0951-7715/12/5/314
[9] Morrison T.P., Parkes E.J.: The N-loop soliton solution of the modified Vakhnenko equation (a new nonlinear evolution equation). Chaos, Solitons Fractals 16, 13–26 (2003) · Zbl 1048.35104 · doi:10.1016/S0960-0779(02)00314-4
[10] Parkes E.J.: Explicit solutions of the reduced Ostrovsky equation. Chaos, Solitons Fractals 31, 602–610 (2007) · Zbl 1138.35399 · doi:10.1016/j.chaos.2005.10.028
[11] Rosenau P., Hyman J.M.: Compactons: solitons with finite wavelength. Phys. Rev. Lett. 70, 564–567 (1993) · Zbl 0952.35502 · doi:10.1103/PhysRevLett.70.564
[12] Rosenau P.: On nonanalytic solitary waves formed by a nonlinear dispersion. Phys. Lett. A 230, 305–318 (1997) · Zbl 1052.35511 · doi:10.1016/S0375-9601(97)00241-7
[13] Rosenau P.: Compact and noncompact dispersive structures. Phys. Lett. A 275(3), 193–203 (2000) · Zbl 1115.35365 · doi:10.1016/S0375-9601(00)00577-6
[14] Sakovich A., Sakovich S.: Solitary wave solutions of the short pulse equation. J. Phys. A Math. Gen. 39, L361–367 (2006) · Zbl 1092.81531 · doi:10.1088/0305-4470/39/22/L03
[15] Stepanyants Y.A.: On stationary solutions of the reduced Ostrovsky equation: periodic wave, compactons and compound solitons. Chaos Solitons Fractals 28, 193–204 (2006) · Zbl 1088.35531 · doi:10.1016/j.chaos.2005.05.020
[16] Tzirtzilakis E., Marinakis V., Apokis C., Bountis T.: Soliton-like solutions of higher order wave equations of the Korteweg-de-Vries Type. J. Math. Phys. 43(12), 6151–6161 (2002) · Zbl 1060.35127 · doi:10.1063/1.1514387
[17] Tzirtzilakis E., Xenos M., Marinakis V., Bountis T.: Interactions and stability of solitary waves in shallow water. Chaos, Solitons Fractals 14, 87–95 (2002) · Zbl 1068.76011 · doi:10.1016/S0960-0779(01)00211-9
[18] Vakhnenko V.O.: High-frequency soliton-like waves in a relaxing medium. J. Math. Phys. 40, 2011–2020 (1999) · Zbl 0946.35094 · doi:10.1063/1.532847
[19] Vakhnenko V.O., Parkes E.J.: The two loop soliton solution of the Vakhnenko equation. Nonlinearity 11, 1457–1464 (1998) · Zbl 0914.35115 · doi:10.1088/0951-7715/11/6/001
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