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New peakon, solitary wave and periodic wave solutions for the modified Camassa-Holm equation. (English) Zbl 1186.34005

Summary: An independent variable transformation is introduced to solve the modified Camassa-Holm equation by using bifurcation theory and the method of phase portrait analysis. Some peakons, solitary waves and periodic waves are found and their exact parametric representations in explicit form and in implicit form are obtained.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
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[1] Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[2] Johnson, R. S., Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455, 63-82 (2002) · Zbl 1037.76006
[3] Ionescu-Kruse, D., Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14, 303-312 (2007) · Zbl 1157.76005
[4] Constantin, A.; Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192, 165-186 (2009) · Zbl 1169.76010
[5] Constantin, A., The trajectories of particles in stokes waves, Invent. Math., 166, 523-535 (2006) · Zbl 1108.76013
[6] Constantin, A.; Escher, J., Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44, 423-431 (2007) · Zbl 1126.76012
[7] Parker, A., On the Camassa-Holm equation and a direct method of solution. II. Soliton solutions, Proc. R. Soc. London A, 461, 3611-3632 (2005) · Zbl 1370.35236
[8] Johnson, R. S., On solutions of the Camassa-Holm equation, Proc. R. Soc. London A, 459, 1687-1708 (2003) · Zbl 1039.76006
[9] Beals, R.; Sattinger, D.; Szmigielski, J., Multi-peakons and a theorem of Stieltjes, Inverse Problems, 15, L1-L4 (1999) · Zbl 0923.35154
[10] Constantin, A.; Strauss, W., Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12, 415-422 (2002) · Zbl 1022.35053
[11] Constantin, A., On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155, 352-363 (1998) · Zbl 0907.35009
[12] Constantin, A.; Gerdjikov, V.; Ivanov, R., Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22, 2197-2207 (2006) · Zbl 1105.37044
[13] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025
[14] Constantin, A., Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier, 50, 321-362 (2000) · Zbl 0944.35062
[15] Constantin, A.; Kolev, B., Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78, 787-804 (2003) · Zbl 1037.37032
[16] Constantin, A.; Kappeler, T.; Kolev, B.; Topalov, P., On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31, 155-180 (2007) · Zbl 1121.35111
[17] Kolev, B., Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst., 19, 555-574 (2007) · Zbl 1139.53040
[18] Lakshmanan, M., Integrable nonlinear wave equations and possible connections to tsunami dynamics, (Tsunami and Nonlinear Waves (2007), Springer: Springer Berlin), 31-49 · Zbl 1310.76044
[19] Constantin, A.; Johnson, R. S., Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res., 40, 175-211 (2008) · Zbl 1135.76007
[20] Wazwaz, A. M., Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations, Phys. Lett. A, 352, 500-504 (2006) · Zbl 1187.35199
[21] Wazwaz, A. M., 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
[22] Liu, Z. R.; Ouyang, Z. Y., A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations, Phys. Lett. A, 366, 377-381 (2007) · Zbl 1203.35234
[23] Wang, Q. D.; Tang, M. Y., New exact solutions for two nonlinear equations, Phys. Lett. A, 372, 2995-3000 (2008) · Zbl 1220.37069
[24] Zhang, B. G.; Li, S. Y.; Liu, Z. R., Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations, Phys. Lett. A, 372, 1867-1872 (2008) · Zbl 1220.34010
[25] Yusufoğlu, E., New solitonary solutions for modified forms of DP and CH equations using Exp-function method, Chaos Solitons Fractals (2007) · Zbl 1197.65115
[26] Rui, W. G.; He, B.; Xie, S. L.; Long, Y., Application of the integral bifurcation method for solving modified Camassa-Holm and Degasperis-Procesi equations, Nonlinear Anal. (2009) · Zbl 1172.35499
[27] He, B.; Li, J. B.; Long, Y.; Rui, W. G., Bifurcations of travelling wave solutions for a variant of Camassa-Holm equation, Nonlinear Anal. RWA, 9, 222-232 (2008) · Zbl 1185.35217
[28] He, B.; Long, Y.; Rui, W. G., New exact bounded travelling wave solutions for the Zhiber-Shabat equation, Nonlinear Anal. (2009) · Zbl 1177.34054
[29] Byrd, P. F.; Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Physicists (1971), Springer-Verlag · Zbl 0213.16602
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