Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients


Sheng Zhang - School of Mathematics and Physics, Bohai University, Jinzhou 121013, China.
Zhaoyu Wang - School of Mathematics and Physics, Bohai University, Jinzhou 121013, China.


In this paper, Whitham–Broer–Kaup (WBK) equations with time-dependent coefficients are exactly solved through Hirota’s bilinear method. To be specific, the WBK equations are first reduced into a system of variable-coefficient Ablowitz–Kaup– Newell–Segur (AKNS) equations. With the help of the AKNS equations, bilinear forms of the WBK equations are then given. Based on a special case of the bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions and the uniform formulae of n-soliton solutions are finally obtained. It is graphically shown that the dynamical evolutions of the obtained one-, two- and three-soliton solutions possess time-varying amplitudes in the process of propagations.



[1] M. J. Ablowitz, P. A. Clarkson,/ Solitons, nonlinear evolution equations and inverse scattering,/ London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge,/ (1991).
[2] M. Arshad, A. R. Seadawy, D.-C. Lu, J. Wang,/ Travelling wave solutions of Drinfeld-Sokolov-Wilson,/ Whitham-Broer- Kaup and (2+1)-dimensional Broer-Kaup-Kupershmit equations and their applications, Chin. J. Phys.,/ (2017), (In press).
[3] D. Baleanu, B. Agheli, R. Darzi,/ Analysis of the new technique to solution of fractional wave- and heat-like equation,/ Acta Phys. Polon. B,/ 48 (2017), 77–95.
[4] D. Baleanu, B. Kilic, M. Inc,/ The first integral method for Wu-Zhang nonlinear system with time-dependent coefficients,/ Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.,/ 16 (2015), 160–167.
[5] D. Y. Chen,/ Introduction of soliton,/ (Chinese), Science Press, Beijing,/ (2006).
[6] S.-H. Chen, P. Grelu, D. Mihalache, F. Baronio,/ Families of rational solutions of the Kadomtsev-Petviashvili equation,/ Romanian Rep. Phys.,/ 68 (2016), 1407–1424.
[7] Y. Chen, Q. Wang,/ Multiple Riccati equations rational expansion method and complexiton solutions of the Whitham-Broer- Kaup equation,/ Phys. Lett. A,/ 347 (2005), 215–227.
[8] Y. Chen, Q. Wang, B. Li,/ A generalized method and general form solutions to the Whitham-Broer-Kaup equation,/ Chaos Solitons Fractals,/ 22 (2004), 675–682.
[9] Y. Chen, Q. Wang, B. Li,/ Elliptic equation rational expansion method and new exact travelling solutions for Whitham- Broer-Kaup equations,/ Chaos Solitons Fractals,/ 26 (2005), 231–246.
[10] D. Y. Chen, X. Y. Zhu, J. B. Zhang, Y. Y. Sun, Y. Shi,/ New soliton solutions to isospectral AKNS equations,/ (Chinese) ; translated from Chinese Ann. Math. Ser. A, 33 (2012), 205–216, Chinese J. Contemp. Math.,/ 33 (2012), 167–176.
[11] S. M. El-Sayed, D. Kaya,/ Exact and numerical traveling wave solutions of Whitham-Broer-Kaup equations,/ Appl. Math. Comput.,/ 167 (2005), 1339–1349.
[12] E.-G. Fan,/ Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems,/ Phys. Lett. A,/ 300 (2002), 243–249.
[13] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura,/ Method for solving the Korteweg-deVries equation,/ Phys. Rev. Lett.,/ 19 (1967), 1095–1097.
[14] J.-H. He, X.-H. Wu,/ Exp-function method for nonlinear wave equations,/ Chaos Solitons Fractals,/ 30 (2006), 700–708.
[15] R. Hirota,/ Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,/ Phys. Rev. Lett.,/ 27 (1971), 1192–1194.
[16] R. Hirota,/ Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons,/ J. Phys. Soc. Japan,/ 33 (1972), 1456–1458.
[17] M. Inc,/ Constructing solitary pattern solutions of the nonlinear dispersive Zakharov-Kuznetsov equation,/ Chaos Solitons Fractals,/ 39 (2009), 109–119.
[18] M. Inç,/ On new exact special solutions of the GNLS(m, n, p, q) equations,/ Modern Phys. Lett. B,/ 24 (2010), 1769–1783.
[19] M. Inç,/ Compact and noncompact structures of a three-dimensional 3DKP(m, n) equation with nonlinear dispersion,/ Appl. Math. Lett.,/ 26 (2013), 437–444.
[20] M. Inç,/ Some special structures for the generalized nonlinear Schrödinger equation with nonlinear dispersion,/ Waves Random Complex Media,/ 23 (2013), 77–88.
[21] M. Inç, E. Ates,/ Optical soliton solutions for generalized NLSE by using Jacobi elliptic functions,/ Optoelectron. Adv. Mat.,/ 9 (2015), 1081–1087.
[22] M. Inç, B. Kilic, D. Baleanu,/ Optical soliton solutions of the pulse propagation generalized equation in parabolic-law media with space-modulated coefficients,/ Optik,/ 127 (2016), 1056–1058.
[23] M. Inç, Z. S. Korpinar, M. M. Al Qurashi, D. Baleanu,/ A new method for approximate solutions of some nonlinear equations: residual power series method,/ Adv. Mech. Eng.,/ 8 (2016), 8 pages.
[24] X.-Y. Jiao, H.-Q. Zhang,/ An extended method and its application to Whitham-Broer-Kaup equation and two-dimensional perturbed KdV equation,/ Appl. Math. Comput.,/ 172 (2006), 664–677.
[25] M. Khalfallah,/ Exact traveling wave solutions of the Boussinesq-Burgers equation,/ Math. Comput. Modelling,/ 49 (2009), 666–671.
[26] D. Kumar, J. Singh, D. Baleanu,/ A hybrid computational approach for Klein-Gordon equations on Cantor sets,/ Nonlinear Dynam.,/ 87 (2017), 511–517.
[27] G.-D. Lin, Y.-T. Gao, L. Wang, D.-X. Meng, X. Yu,/ Elastic-inelastic-interaction coexistence and double Wronskian solutions for the Whitham-Broer-Kaup shallow-water-wave model,/ Commun. Nonlinear Sci. Numer. Simul.,/ 16 (2011), 3090–3096.
[28] Y.-B. Liu, A. S. Fokas, D. Mihalache, J.-S. He,/ Parallel line rogue waves of the third-type Davey-Stewartson equation,/ Romanian Rep. Phys.,/ 68 (2016), 1425–1446.
[29] Q. P. Liu, X.-B. Hu, M.-X. Zhang,/ Supersymmetric modified Korteweg-de Vries equation: bilinear approach,/ Nonlinearity,/ 18 (2005), 1597–1603.
[30] Y. Liu, X.-Q. Liu,/ Exact solutions of Whitham-Broer-Kaup equations with variable coefficients,/ Acta Phys. Sin.,/ 63 (2014), 9 pages.
[31] V. B. Matveev, M. A. Salle,/ Darboux transformations and solitons,/ Springer Series in Nonlinear Dynamics, Springer- Verlag, Berlin,/ (1991).
[32] I. N. McArthur, C. M. Yung,/ Hirota bilinear form for the super-KdV hierarchy,/ Modern Phys. Lett. A,/ 8 (1993), 1739– 1745.
[33] M. R. Miura,/ Bäcklund transformation,/ Springer-Verlag, Berlin,/ (1978).
[34] A. Mohebbi, Z. Asgari, M. Dehghan,/ Numerical solution of nonlinear Jaulent-Miodek and Whitham-Broer-Kaup equations,/ Commun. Nonlinear Sci. Numer. Simul.,/ 17 (2012), 4602–4610.
[35] S. T. Mohyud-Din, A. Yıldırım, G. Demirli,/ Traveling wave solutions of Whitham-Broer-Kaup equations by homotopy perturbation method,/ J. King Saud Univ. Sci.,/ 22 (2010), 173–176.
[36] M. Rafei, H. Daniali,/ Application of the variational iteration method to the Whitham-Broer-Kaup equations,/ Comput. Math. Appl.,/ 54 (2007), 1079–1085.
[37] V. N. Serkin, A. Hasegawa,/ Novel soliton solutions of the nonlinear Schrödinger equation model,/ Phys. Rev. Lett.,/ 85 (2000), 4502–4505.
[38] V. N. Serkin, A. Hasegawa, T. L. Belyaeva,/ Nonautonomous solitons in external potentials,/ Phys. Rev. Lett.,/ 98 (2007), 4 pages.
[39] V. N. Serkin, A. Hasegawa, T. L. Belyaeva,/ Nonautonomous matter-wave solitons near the Feshbach resonance,/ Phys. Rev. A,/ 81 (2010), 19 pages.
[40] J.-W. Shen, W. Xu, Y.-F. Jin,/ Bifurcation method and traveling wave solution to Whitham-Broer-Kaup equation,/ Appl. Math. Comput.,/ 171 (2005), 677–702.
[41] M. Song, J. Cao, X.-L. Guan,/ Application of the bifurcation method to the Whitham-Broer-Kaup-like equations,/ Math. Comput. Modelling,/ 52 (2012), 688–696.
[42] H. Triki, H. Leblond, D. Mihalache,/ Soliton solutions of nonlinear diffusion-reaction-type equations with time-dependent coefficients accounting for long-range diffusion,/ Nonlinear Dynam.,/ 86 (2016), 2115–2126.
[43] H. Triki, A.-M. Wazwaz,/ Soliton solutions of the cubic-quintic nonlinear Schrodinger equation with variable coefficients,/ Romanian J. Phys.,/ 61 (2016), 360–366.
[44] M.-L. Wang,/ Exact solutions for a compound KdV-Burgers equation,/ Phys. Lett. A,/ 213 (1996), 279–287.
[45] A.-M. Wazwaz,/ The Hirota’s bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera Kadomtsev-Petviashvili equation,/ Appl. Math. Comput.,/ 200 (2008), 160–166.
[46] J. Weiss, M. Tabor, G. Carnevale,/ The Painlevé property for partial differential equations,/ J. Math. Phys.,/ 24 (1983), 522–526.
[47] X.-Y. Wen,/ A new integrable lattice hierarchy associated with a discrete \(3 \times 3\) matrix spectral problem: N-fold Darboux transformation and explicit solutions,/ Rep. Math. Phys.,/ 71 (2013), 15–32.
[48] F.-D. Xie, Z.-Y. Yan, H.-Q. Zhang,/ Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations,/ Phys. Lett. A,/ 285 (2001), 76–80.
[49] G.-Q. Xu, Z.-B. Li,/ Exact travelling wave solutions of the Whitham-Broer-Kaup and Broer-Kaup-Kupershmidt equations,/ Chaos Solitons Fractals,/ 24 (2005), 549–556.
[50] S.-W. Xu, K. Porsezian, J.-S. He, Y. Cheng,/ Multi-optical rogue waves of the Maxwell-Bloch equations,/ Romanian Rep. Phys.,/ 68 (2016), 316–340.
[51] Z.-L. Yan, X.-Q. Liu,/ Solitary wave and non-traveling wave solutions to two nonlinear evolution equations,/ Commun. Theor. Phys. (Beijing),/ 44 (2005), 479–482.
[52] Z.-Y. Yan, H.-Q. Zhang,/ New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water,/ Phys. Lett. A,/ 285 (2001), 355–362.
[53] Z.-L. Yan, J.-P. Zhou,/ New explicit solutions of (1 + 1)-dimensional variable-coefficient Broer-Kaup system,/ Commun. Theor. Phys. (Beijing),/ 54 (2010), 965–970.
[54] X.-J. Yang, D. Baleanu, H. M. Srivastava,/ Local fractional integral transforms and their applications,/ Elsevier/Academic Press, Amsterdam,/ (2015).
[55] S. Zhang,/ Application of Exp-function method to a KdV equation with variable coefficients,/ Phys. Lett. A,/ 365 (2007), 448–453.
[56] S. Zhang,/ Exact solutions of a KdV equation with variable coefficients via Exp-function method,/ Nonlinear Dynam.,/ 52 (2008), 11–17.
[57] P. Zhang,/ New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations,/ Appl. Math. Comput.,/ 217 (2010), 1688–1696.
[58] S. Zhang, B. Cai,/ Multi-soliton solutions of a variable-coefficient KdV hierarchy,/ Nonlinear Dynam.,/ 78 (2014), 1593– 1600.
[59] S. Zhang, M.-T. Chen,/ Painlevé integrability and new exact solutions of the (4 + 1)-dimensional Fokas equation,/ Math. Probl. Eng.,/ 2015 (2015), 8 pages.
[60] S. Zhang, M.-T. Chen, W.-Y. Qian,/ Painlevé analysis for a forced Korteveg-de Vries equation arisen in fluid dynamics of internal solitary waves,/ Therm. Sci.,/ 19 (2015), 1223–1226.
[61] S. Zhang, X.-D. Gao,/ Mixed spectral AKNS hierarchy from linear isospectral problem and its exact solutions,/ Open Phys.,/ 13 (2015), 310–322.
[62] S. Zhang, X.-D. Gao,/ Exact N-soliton solutions and dynamics of a new AKNS equation with time-dependent coefficients,/ Nonlinear Dynam.,/ 83 (2016), 1043–1052.
[63] S. Zhang, D. Liu,/ Multisoliton solutions of a (2 + 1)-dimensional variable-coefficient Toda lattice equation via Hirota’s bilinear method,/ Canad. J. Phys.,/ 92 (2014), 184–190.
[64] S. Zhang, D.-D. Liu,/ The third kind of Darboux transformation and multisoliton solutions for generalized Broer-Kaup equations,/ Turkish J. Phys.,/ 39 (2015), 165–177.
[65] S. Zhang, C. Tian, W.-Y. Qian,/ Bilinearization and new multisoliton solutions for the (4 + 1)-dimensional Fokas equation,/ Pramana,/ 86 (2016), 1259–1267.
[66] S. Zhang, D. Wang,/ Variable-coefficient nonisospectral Toda lattice hierarchy and its exact solutions,/ Pramana,/ 85 (2015), 1143–1156.
[67] S. Zhang, T.-C. Xia,/ A generalized F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equations,/ Appl. Math. Comput.,/ 183 (2006), 1190–1200.
[68] S. Zhang, T.-C. Xia,/ A generalized auxiliary equation method and its application to (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equations,/ J. Phys. A,/ 40 (2007), 227–248.
[69] S. Zhang, B. Xu, H.-Q. Zhang,/ Exact solutions of a KdV equation hierarchy with variable coefficients,/ Int. J. Comput. Math.,/ 91 (2014), 1601–1616.
[70] S. Zhang, H.-Q. Zhang,/ An Exp-function method for a new N-soliton solutions with arbitrary functions of a (2 + 1)- dimensional vcBK system,/ Comput. Math. Appl.,/ 61 (2011), 1923–1930.
[71] S. Zhang, H.-Q. Zhang,/ Fractional sub-equation method and its applications to nonlinear fractional PDEs,/ Phys. Lett. A,/ 375 (2011), 1069–1073.
[72] S. Zhang, L.-Y. Zhang,/ Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients,/ Open Phys.,/ 14 (2016), 69–75.