Abstract
Based on the corresponding bilinear forms, we obtain one-, two- and three-soliton solutions of a new AKNS equation with time-dependent coefficients by using Hirota’s bilinear method. From these obtained solutions, uniform formulae of exact N-soliton solutions of the AKNS equation are summarized. It is graphically shown that the dynamical evolutions of such soliton solutions with time-dependent functions of the AKNS equation possess time-varying speeds and amplitudes in the process of propagations.
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Drazin, P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge University Press, New York (1989)
Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)
Miurs, M.R.: Bäcklund Transformation. Springer, Berlin (1978)
Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, 650–654 (1992)
Wang, M.L.: Exact solutions for a compound KdV–Burgers equation. Phys. Lett. A 213, 279–287 (1996)
Liu, S.K., Fu, Z.T., Liu, S.D., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289, 69–74 (2001)
Fan, E.G.: Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems. Phys. Lett. A 300, 243–249 (2002)
Fan, E.G.: Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos Solitons Fractals 16, 819–839 (2003)
Yomba, E.: The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer–Kaup–Kupershmidt equation. Chaos Solitons Fractals 27, 187–196 (2007)
Zhang, S., Xia, T.C.: A generalized auxiliary equation method and its application to (2+1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equations. J. Phys. A Math. Theor. 40, 227–248 (2007)
He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700–708 (2006)
Zhang, S.: Exact solutions of a KdV equation with variable coefficients via Exp-function method. Nonlin. Dyn. 52, 11–17 (2008)
Ma, W.X., Lee, J.H.: A transformed rational function method and exact solutions to 3+1 dimensional Jimbo–Miwa equation. Chaos Solitons Fractals 42, 1356–1363 (2009)
Dai, C.Q., Wang, Y.Y., Tian, Q., Zhang, J.F.: The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation. Ann. Phys. 327, 512–521 (2012)
Antonova, M., Biswas, A.: Adiabatic parameter dynamics of perturbed solitary waves. Commun. Nonlin. Sci. Numer. Simul. 14, 734–748 (2009)
Mirzazadeh, M., Eslami, M., Biswas, A.: 1-Soliton solution to KdV6 equation. Nonlin. Dyn. 80, 387–396 (2015)
Bhrawy, A.H., Abdelkawy, M.A., Biswas, A.: Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method. Commun. Nonlin. Sci. Numer. Simul. 18, 915–925 (2013)
Bhrawy, A.H., Biswas, A., Javidi, M., Ma, W.X., Pinar, Z., Yildirim, A.: New solutions for (1+1)-dimensional and (2+1)-dimensional Kaup–Kupershmidt equations. Results Math. 63, 675–686 (2013)
Bhrawy, A.H., Abdelkawy, M.A., Kumar, S., Biswas, A.: Solitons and other solutions to Kadomtsev–Petviashvili equation of b-type. Rom. J. Phys. 58, 729–748 (2013)
Ebadi, G., Mojaver, A., Triki, H., Yildirim, A., Biswas, A.: Topological solitons and other solutions to the (3+1)-dimensional extended Kadomtsev–Petviashvili equation with power law nonlinearity. Rom. J. Phys. 58, 3–14 (2013)
Triki, H., Kara, A.H., Bhrawy, A., Biswas, A.: Soliton solution and conservation law of Gear–Grimshaw model for shallow water waves. Acta Phys. Pol. A 125, 1099–1106 (2014)
Triki, H., Mirzazadeh, M., Bhrawy, A.H., Razborova, P., Biswas, A.: Soliton and other solutions to long-wave short-wave interaction equation. Rom. J. Phys. 60, 72–86 (2015)
Razborova, P., Moraru, L., Biswas, A.: Perturbation of dispersive shallow water waves with Rosenau–KdV–RLW equation with power law nonlinearity. Rom. J. Phys. 59, 658–676 (2014)
Razborova, P., Triki, H., Biswas, A.: Perturbation of dispersive shallow water waves. Ocean Eng. 63, 1–7 (2013)
Dai, C.Q., Wang, X.G., Zhou, G.Q.: Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A 89, 013834 (2014)
Fan, E.G., Chow, K.W., Li, J.H.: On doubly periodic standing wave solutions of the coupled higgs field equation. Stud. Appl. Math. 128, 86–105 (2012)
Zayed, E.M.E., Abdelaziz, M.A.M.: An extended (G\(^{\prime }\)/G)- expansion method and its applications to the (2+1)-dimensional nonlinear evolution equations. WSEAS Trans. Math. 11, 1039–1047 (2012)
Wang, Y.L.: Variable-coefficient simplest equation method for solving nonlinear evolution equations in mathematical physics. WSEAS Trans. Math. 12, 512–520 (2013)
Kamenov, O.Y.: New periodic exact solutions of the Kuramoto–Sivashinsky evolution equation. WSEAS Trans. Math. 13, 345–352 (2014)
Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)
Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, New York (2004)
Balashov, M.V.: A property of the ansatz of Hirota’s method for quasilinear parabolic equations. Math. Notes 71, 339–354 (2002)
Zuo, D.W., Gao, Y.T., Meng, G.Q., Shen, Y.J., Yu, X.: Multi-soliton solutions for the three-coupled KdV equations engendered by the Neumann system. Nonlin. Dyn. 75, 701–708 (2014)
Mcarthur, I., Yung, C.M.: Hirota bilinear form for the super-KdV hierarchy. Mod. Phys. Lett. A 8, 1739–1745 (1993)
Liu, Q.P., Hu, X.B., Zhang, M.X.: Supersymmetric modified Korteweg–de Vries equation: bilinear approach. Nonlinearity 18, 1597–1603 (2005)
Chen, D.Y.: Introduction to Soliton. Science Press, Beijing (2006)
Wazwaz, A.M.: 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, 160–166 (2008)
Zhang, S., Liu, D.: Multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation via Hirota’s bilinear method. Can. J. Phys. 92, 184–190 (2014)
Zhang, S., Cai, B.: Multi-soliton solutions of a variable-coefficient KdV hierarchy. Nonlin. Dyn. 78, 1593–1600 (2014)
Liu, Y., Gao, Y.T., Sun, Z.Y., Yu, X.: Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary waves. Nonlin. Dyn. 66, 575–587 (2011)
Zabusky, N.J., Kruskal, M.D.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)
Chen, H.H., Liu, C.S.: Solitons in nonuniform media. Phys. Rev. Lett. 37, 693–697 (1976)
Hirota, R., Satsuma, J.: \(N\)-soliton solutions of the K-dV equation with loss and nonuniformity terms. J. Phys. Soc. Jpn. Lett. 41, 2141–2142 (1976)
Calogero, F., Degasperis, A.: Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron. Lett. Nuovo Cimento 16, 425–433 (1976)
Chen, H.H., Liu, C.S.: Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities. Phys. Fluids. 21, 377–380 (1978)
Serkin, V.N., Hasegawa, A.: Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85, 4502–4505 (2000)
Serkin, V.N., Belyaeva, T.L.: The Lax representation in the problem of soliton management. Quant. Electron. 31, 1007–1015 (2001)
Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98, 074102 (2007)
Serkin, A., Hasegawa, V.N., Belyaeva, T.L.: Solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomous solitons. J. Mod. Opt. 57, 1456–1472 (2010)
Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous matter-wave solitons near the Feshbach resonance. Phys. Rev. A 81, 023610 (2010)
Acknowledgments
We would like to express our sincerest thanks to the referee for the valuable suggestions and comments which lead to further improvement of our original manuscript. This work was supported by the Natural Science Foundation of Liaoning Province of China (L2012404), the Ph.D. Start-up Funds of Liaoning Province of China (20141137) and Bohai University (bsqd2013025), the Liaoning BaiQianWan Talents Program (2013921055) and the Natural Science Foundation of China (11371071).
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Zhang, S., Gao, X. Exact \(\varvec{N}\)-soliton solutions and dynamics of a new AKNS equation with time-dependent coefficients. Nonlinear Dyn 83, 1043–1052 (2016). https://doi.org/10.1007/s11071-015-2386-5
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DOI: https://doi.org/10.1007/s11071-015-2386-5