Explicit compacton and generalized kink wave solutions for a CH-DP equation

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Abstract

This study aimed to investigate the traveling solutions of the Camassa-Holm and Degasperis-Procesi (CH-DP) equation. By using the qualitative theory of dynamical systems and integrating along special phase orbits, we present exact explicit expressions of compacton and generalized kink wave solutions to the CH-DP equation. Our results will enrich the previous literature and help to understand the propagation of nonlinear waves.

Keywords

CH-DP equation / compacton / generalized kink / exact explicit expression

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References

[1] C.S.Gardner,J.M.Greene,M.D.Kruskal,R.M.Miura, Method for solving the Korteweg-de Vries equation, Phys.Rev.Lett., 19(1967), 1095-1097. https://doi.org/10.1103/PhysRevLett.19.1095

[2] M.R.Miura, Bäcklund Transformations ,Springer-Verlag,Berlin, 1978.

[3] G.T.Liu,T.Y.Fan, New applications of developed Jacobi elliptic function expansion methods, Phys.Lett.A, 345(2005), 161-166. https://doi.org/10.1016/j.physleta.2005.07.034

[4] S.K.Liu,Z.T.Fu,S.D.Liu,Q.Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys.Lett.A, 289(2001), 69-74. https://doi.org/10.1016/S0375-9601(01)00580-1

[5] M.L.Wang,X.Z.Li,J.L.Zhang, The G′ G-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , Phys.Lett.A, 372(2008), 417-423. https://doi.org/10.1016/j.physleta.2007.07.051

[6] M.Song,Y.L.Ge, Application of the G′ G-expansion method to (3+1)-dimensional nonlinear evolution equations , Comput.Math.Appl., 60(2010), 1220-1227. https://doi.org/10.1016/j.camwa.2010.05.045

[7] W.G.Rui, The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation, Nonlinear Dyn., 76(2014), 1529-1542. https://doi.org/10.1007/s11071-013-1226-8

[8] R.Camassa,D.D.Holm, An integrable shallow water equation with peaked solitons, Phys.Rev.Lett., 71(1993), 1661-1664. https://doi.org/10.1103/PhysRevLett.71.1661

[9] Z.R.Liu,R.Q.Wang,Z.J.Jing, Peaked wave solutions of Camassa-Holm equation, Chaos Solitons Fractals, 19(2004), 77-92. https://doi.org/10.1016/S0960-0779(03)00082-1

[10] Y.A.Li,P.J.Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J.Differ.Equations, 162(2000), 27-63. https://doi.org/10.1006/jdeq.1999.3683

[11] Z.Y.Yin, Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation, Commun.Pure Appl.Anal., 3(2004), 501-508. https://doi.org/10.3934/cpaa.2004.3.501

[12] Y.Liu, Global existence and blow-up solutions for a nonlinear shallow water equation, Math.Ann., 335(2006), 717-735. https://doi.org/10.1007/s00208-006-0768-1

[13] A.Degasperis,M.Procesi, Asymptotic integrability,in Symmetry and Perturbation Theory (eds.A.Degasperis,G.Gaeta),World Scientific,Singapore,(1999), 23-37. https://doi.org/10.1142/9789812833037

[14] P.Rosenau,J.M.Hyman, Compactons:Solitons with finite wavelength, Phys.Rev.Lett., 70(1993), 564-567. https://doi.org/10.1103/PhysRevLett.70.564

[15] Z.R.Liu,Q.X.Li,Q.M.Lin, New bounded traveling waves of Camassa-Holm equation, Int.J.Bifurca.Chaos., 14(2004), 3541-3556. https://doi.org/10.1142/S0218127404011521

[16] H.R.Dullin,G.A.Gottwald,D.D.Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys.Rev.Lett., 87(2001), 194501-194504. https://doi.org/10.1103/PhysRevLett.87.194501

[17] B.He,J.B.Li,Y.Long,W.G.Rui, Bifurcations of travelling wave solutions for a variant of Camassa-Holm equation, Nonlinear Anal.Real World Appl., 9(2008), 222-232. https://doi.org/10.1016/j.nonrwa.2006.10.001

[18] J.B.Li,Y.Zhang, Exact loop solutions,cusp solutions,solitary wave solutions and periodic wave solutions for the special CH-DP equation, Nonlinear Anal.Real World Appl., 10(2009), 2502-2507. https://doi.org/10.1016/j.nonrwa.2008.05.006

[19] S.L.Xie,Q.Lin,B.Gao, Periodic and solitary travelling-wave solutions of a CH-DP equation, Commun.Nonlinear Sci.Numer.Simul., 16(2011), 3941-3948. https://doi.org/10.1016/j.cnsns.2011.01.023

[20] S.L.Xie,L.Wang, Compacton and generalized kink wave solutions of the CH-DP equation, Appl.Math.Comput., 215(2010), 4028-4039. https://doi.org/10.1016/j.amc.2009.12.010

[21] X.M.Lyu,Q.Ge, Numerical study of bell-shaped solitons solutions for a generalized modified CH-DP equation, Electron.Res.Arch., 33(2025), 4603-4624. https://doi.org/10.3934/era.2025207

[22] B.L.Guo,Z.R.Liu, Peaked wave solutions of CH-γ equation, Sci.China Ser.A, 46(2003), 696-709. https://doi.org/10.1007/BF02942241

[23] B.L.Guo,Z.R.Liu, Two new types of bounded waves of CH-γ equation, Sci.China Ser.A, 48(2005), 1618-1630. https://doi.org/10.1360/04ys0205

[24] M.Y.Tang,W.L.Zhang, Four types of bounded wave solutions of CH-γ equation, Sci.China Ser.A Math., 50(2007), 132-152. https://doi.org/10.1007/s11425-007-2042-8

[25] J.B.Li,Z.R.Liu, Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Appl.Math.Model., 25(2000), 41-56. https://doi.org/10.1016/S0307-904X(00)00031-7

[26] S.Y.Li,M.Song, Compacton-like wave and kink-like wave solutions of the generalized KP-MEW (2,2) equation, Phys.Scr., 89(2014), 035202. https://doi.org/10.1088/0031-8949/89/03/035202

[27] S.Y.Li,Z.R.Liu, Kink-like wave and compacton-like wave solutions for generalized KdV equation, Nonlinear Dyn., 79(2015), 903-918. https://doi.org/10.1007/s11071-014-1710-9

[28] Y.Li,S.Y.Li,R.Y.Wei, Bifurcation analysis and solitary-like wave solutions for extended (2+1)-dimensional Konopelchenko-Dubrovsky equations, Nonlinear Dyn., 88(2017), 609-622. http://dx.doi.org/10.1007/s11071-016-3264-5

[29] M.Song,B.D.Wang,S.Y.Li, Bifurcation of traveling wave solutions for (1+1)-dimensional resonant nonlinear Schrödinger equation, J.Math.Anal.Appl., 509(2022), 125965. https://doi.org/10.1016/j.jmaa.2021.125965