×

Elastic and inelastic interactions of solitons for a variable-coefficient generalized dispersive water-wave system. (English) Zbl 1297.76039

Summary: Under consideration in this paper is a variable-coefficient generalized dispersive water-wave system which can model the propagation of long weakly nonlinear and weakly dispersive surface waves of variable depth in shallow water. With the aid of symbolic computation and \(N\)-fold Darboux transformation, multi-soliton solutions of the system are obtained. It is found that solitonic interactions could be either the inelastic fusion-fission or elastic ones, depending on different choices of variable coefficients. With the parameters periodically varying with time, waves change their shapes while the fusion-fission interactions also coexist and are periodic.

MSC:

76B25 Solitary waves for incompressible inviscid fluids
35Q51 Soliton equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993) · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661
[2] Wang, L., Gao, Y.T., Meng, D.X., Gai, X.L., Xu, P.B.: Soliton-shape-preserving and soliton-complex interactions for a (1+1)-dimensional nonlinear dispersive-wave system in shallow water. Nonlinear Dyn. 66, 161–168 (2010) · Zbl 1392.35269 · doi:10.1007/s11071-010-9918-9
[3] Li, Y.S., Ma, W.X., Zhang, J.E.: Darboux transformations of classical Boussinesq system and its new solutions. Phys. Lett. A 275, 60–66 (2000) · Zbl 1115.35329 · doi:10.1016/S0375-9601(00)00583-1
[4] Lin, G.D., Gao, Y.T., Gai, X.L., Meng, D.X.: Extended double Wronskian solutions to the Whitham–Broer–Kaup equations in shallow water. Nonlinear Dyn. 64, 197–206 (2010) · Zbl 1284.35344 · doi:10.1007/s11071-010-9857-5
[5] Yu, X., Gao, Y.T., Sun, Z.Y., Liu, Y.: Wronskian solutions and integrability for a generalized variable-coefficient forced Korteweg–de Vries equation in fluids. Nonlinear Dyn. (2011). doi: 10.1007/s11071-011-0044-0 · Zbl 1356.35212
[6] Tian, B., Gao, Y.T.: Spherical Kadomtsev–Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation. Phys. Lett. A 340, 243–250 (2005) · Zbl 1145.35451 · doi:10.1016/j.physleta.2005.03.035
[7] Tian, B., Gao, Y.T.: On the solitonic structures of the cylindrical dust-acoustic and dust-ion-acoustic waves with symbolic computation. Phys. Lett. A 340, 449–455 (2005) · Zbl 1145.35452 · doi:10.1016/j.physleta.2005.03.082
[8] Tian, B., Gao, Y.T.: Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas. Phys. Lett. A 362, 283–288 (2007) · Zbl 1197.82028 · doi:10.1016/j.physleta.2006.10.094
[9] Lü, X., Tian, B., Zhang, H.Q., Xu, T., Li, H.: Generalized (2+1)-dimensional Gardner model: bilinear equations, Bäcklund transformation, Lax representation and interaction mechanisms. Nonlinear Dyn. (2011). doi: 10.1007/s11071-011-0145-9 · Zbl 1427.74156 · doi:10.1007/s11071-010-9906-0
[10] Kupershmidt, B.A.: Mathematics of Dispersive Water Waves. Commun. Math. Phys. 99, 51–73 (1985) · Zbl 1093.37511 · doi:10.1007/BF01466593
[11] Zhang, J.E., Li, Y.S.: Bidirectional solitons on water. Phys. Rev. E 67, 016306 (2003) · Zbl 1048.35107
[12] Meng, D.X., Gao, Y.T., Wang, L., Gai, X.L., Lin, G.D.: Interactions of solitons in a variable-coefficient generalized Boussinesq system in shallow water. Phys. Scr. 82, 045012 (2010) · Zbl 1202.76027
[13] Singh, K., Gupta, R.K.: Exact solutions of a variant Boussinesq system. Int. J. Eng. Sci. 44, 1256–1268 (2006) · Zbl 1213.35362 · doi:10.1016/j.ijengsci.2006.07.009
[14] Zhao, M., Teng, B., Cheng, L.: A new form of the generalized Boussinesq equations for varying water depth. Ocean Eng. 31, 2047–2072 (2004) · doi:10.1016/j.oceaneng.2004.03.010
[15] Meng, D.X., Gao, Y.T., Wang, L., Xu, P.B.: Painleve property, Lax pair and soliton-interaction for a variable-coefficient generalized dispersive water-wave system (2011, in preparation)
[16] Lin, J., Ren, B., Li, H.M., Li, Y.S.: Soliton solutions for two nonlinear partial differential equations using a Darboux transformation of the Lax pairs. Phys. Rev. E 77, 036605 (2008)
[17] Kaup, D.J.: A higher-order water wave equation and the method for solving it. Prog. Theor. Phys. 54, 396–408 (1975) · Zbl 1079.37514 · doi:10.1143/PTP.54.396
[18] Clarkson, P.A., Ludlow, D.K.: Symmetry reductions, exact solutions and Painlevé analysis of a generalized Boussinesq equation. J. Math. Anal. Appl. 186, 132–155 (1994) · Zbl 0810.35083 · doi:10.1006/jmaa.1994.1290
[19] Weiss, J., Tabor, M., Carnevale, G.: The Painlevé property for partial differential equations. J. Math. Phys. 24, 522–526 (1983) · Zbl 0514.35083 · doi:10.1063/1.525721
[20] Fan, E.G.: Auto-Bäklund transformation and similarity reductions for general variable coefficient KdV equations. Phys. Lett. A 294, 26–30 (2002) · Zbl 0981.35064 · doi:10.1016/S0375-9601(02)00033-6
[21] Tian, B., Gao, Y.T.: Spherical nebulons and Bäcklund transformation for a space or laboratory un-magnetized dusty plasma with symbolic computation. Eur. Phys. J. D 33, 59–65 (2005) · doi:10.1140/epjd/e2005-00036-6
[22] Tian, B., Gao, Y.T.: Cylindrical nebulons, symbolic computation and Bäcklund transformation for the cosmic dust acoustic waves. Phys. Plasmas 12, 070703 (2005)
[23] Tian, B., Gao, Y.T.: Comment on ”Exact solutions of cylindrical and spherical dust ion acoustic waves” [Phys. Plasmas 10, 4162 (2003)]. Phys. Plasmas 12, 054701 (2005)
[24] Lü, X., Zhu, H.W., Meng, X.H., Yang, Z.C., Tian, B.: Soliton solutions and a Bäklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications. J. Math. Anal. Appl. 336, 1305–1315 (2007) · Zbl 1128.35385 · doi:10.1016/j.jmaa.2007.03.017
[25] Liu, W.J., Tian, B., Zhang, H.Q., Li, L.L., Xue, Y.S.: Soliton interaction in the higher-order nonlinear Schrödinger equation investigated with Hirota’ bilinear method. Phys. Rev. E 77, 066605 (2008)
[26] Liu, W.J., Tian, B., Zhang, H.Q.: Types of solutions of the variable-coefficient nonlinear Schrödinger equation with symbolic computation. Phys. Rev. E 78, 066613 (2008)
[27] Liu, W.J., Tian, B., Zhang, H.Q., Xu, T., Li, H.: Solitary wave pulses in optical fibers with normal dispersion and higher-order effects. Phys. Rev. A 79, 063810 (2009)
[28] Xu, T., Tian, B., Li, L.L., Lü, X., Zhang, C.: Dynamics of Alfvén solitons in inhomogeneous plasmas. Phys. Plasmas 15, 102307 (2008)
[29] Xu, T., Tian, B.: Bright N-soliton solutions in terms of the triple Wronskian for the coupled nonlinear Schrödinger equations in optical fibers. J. Phys. A 43, 245205 (2010) · Zbl 1191.81119
[30] Xu, T., Tian, B.: An extension of the Wronskian technique for the multicomponent Wronskian solution to the vector nonlinear Schrödinger equation. J. Math. Phys. 51, 033504 (2010) · Zbl 1309.35146
[31] Zhang, H.Q., Tian, B., Lü, X., Li, H., Meng, X.H.: Soliton interaction in the coupled mixed derivative nonlinear Schrödinger equations. Phys. Lett. A 373, 4315–4321 (2009) · Zbl 1234.35259 · doi:10.1016/j.physleta.2009.09.010
[32] Zhang, H.Q., Tian, B., Lü, X., Li, H., Meng, X.H.: Soliton interaction in the coupled mixed derivative nonlinear Schrödinger equations. Phys. Lett. A 373, 4315–4321 (2009) · Zbl 1234.35259 · doi:10.1016/j.physleta.2009.09.010
[33] Zhang, H.Q., Tian, B., Xu, T., Li, H., Zhang, C., Zhang, H.: Lax pair and Darboux transformation for multi-component modified Korteweg–de Vries equations. J. Phys. A 41, 355210 (2008) · Zbl 1144.82007
[34] Zhang, H.Q., Tian, B., Meng, X.H., Lü, X., Liu, W.J.: Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation. Eur. Phys. J. B 72, 233–239 (2009) · Zbl 1188.35184 · doi:10.1140/epjb/e2009-00356-3
[35] Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993) · Zbl 0785.58003
[36] Karpman, V.I.: Radiation by weakly nonlinear shallow-water solitons due to higher-order dispersion. Phys. Rev. E 58, 5070–5080 (1998) · doi:10.1103/PhysRevE.58.5070
[37] Mingaleev, S.F., Gaididei, Y.B.: Solitons in anharmonic chains with power-law long-range interactions. Phys. Rev. E 58, 3833 (1998) · doi:10.1103/PhysRevE.58.3833
[38] Huang, G.X., Velarde, M.G., Makarov, V.A.: Dark solitons and their head-on collisions in Bose–Einstein condensates. Phys. Rev. A 64, 013617 (2001)
[39] Demiray, H.: Interaction of nonlinear waves governed by Boussinesq equation. Chaos Solitons Fractals 30, 1185–1189 (2006) · Zbl 1142.35564 · doi:10.1016/j.chaos.2005.08.185
[40] Lü, X., Tian, B., Xu, T., Cai, K.J., Liu, W.J.: Analytical study of the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential in quasi-one-dimensional Bose-Einstein condensates via symbolic computation. Ann. Phys. 323, 2554 (2008) · Zbl 1155.35454 · doi:10.1016/j.aop.2008.04.008
[41] Lü, X., Zhu, H.W., Yao, Z.Z., Meng, X.H., Zhang, C., Zhang, C.Y., Tian, B.: Multisoliton solutions in terms of double Wronskian determinant for a generalized variable-coefficient nonlinear Schrödinger equation from plasma physics, arterial mechanics, fluid dynamics and optical communications. Ann. Phys. 323, 1947 (2008) · Zbl 1155.35455 · doi:10.1016/j.aop.2007.10.007
[42] Lü, X., Li, J., Zhang, H.Q., Xu, T., Tian, B.: Integrability aspects with optical solitons of a generalized variable-coefficient N-coupled higher order nonlinear Schrödinger system from inhomogeneous optical fibers. J. Math. Phys. 51, 043511 (2010) · Zbl 1310.35219
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.