Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients
-
2725
Downloads
-
5189
Views
Authors
Sheng Zhang
- School of Mathematics and Physics, Bohai University, Jinzhou 121013, China.
Zhaoyu Wang
- School of Mathematics and Physics, Bohai University, Jinzhou 121013, China.
Abstract
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.
Share and Cite
ISRP Style
Sheng Zhang, Zhaoyu Wang, Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 5, 2324--2339
AMA Style
Zhang Sheng, Wang Zhaoyu, Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients. J. Nonlinear Sci. Appl. (2017); 10(5):2324--2339
Chicago/Turabian Style
Zhang, Sheng, Wang, Zhaoyu. "Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients." Journal of Nonlinear Sciences and Applications, 10, no. 5 (2017): 2324--2339
Keywords
- Bilinear form
- soliton solution
- WKB equations with time-dependent coefficients
- Hirota’s bilinear method.
MSC
References
-
[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.