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Bright and dark soliton solutions and Bäcklund transformation for the Eckhaus-Kundu equation with the cubic-quintic nonlinearity. (English) Zbl 1328.37055
Summary: With symbolic computation, the Eckhaus-Kundu equation which appears in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics, is studied via the Hirota method. By virtue of the dependent variable transformation, the bilinear form is obtained. Bilinear Bäcklund transformation is given with the help of exchange formulae and the corresponding one-soliton solution is derived. Bright and dark \( N\)-soliton solutions are obtained. Propagation and interaction of the bright and dark solitons are discussed analytically and graphically. Interactions of the two solitons are presented. Bound state of the two solitons can be suppressed via the choice of parameters.

37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35C07 Traveling wave solutions
35Q40 PDEs in connection with quantum mechanics
Full Text: DOI
[1] Pelap, F. B.; Faye, M. M., Solitonlike excitations in a one-dimensional electrical transmission line, J. Math. Phys., 46, 033502-033511, (2005) · Zbl 1067.94021
[2] Zhao, X. Q.; Tang, D. B.; Wang, L. M., New soliton-like solutions for KdV equation with variable coefficient, Phys. Lett. A, 346, 288-291, (2005) · Zbl 1195.35275
[3] Karlsson, M.; Kaup, D. J.; Malomed, B. A., Interactions between polarized soliton pulses in optical fibers: exact solutions, Phys. Rev. E, 54, 5802-5808, (1996)
[4] Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G., Quantum inverse scattering method and correlation functions, (1993), Cambridge University Press Cambridge · Zbl 0787.47006
[5] Khater, A. H.; El-Kalaawy, O. H.; Callebaut, D. K., Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron-positron plasma, Phys. Scr., 58, 545-548, (1998)
[6] Estévez, P. G., Darboux transformation and solutions for an equation in 2+1 dimensions, J. Math. Phys., 40, 1406-1419, (1999) · Zbl 0943.35078
[7] Hirota, R.; Ohta, Y., Hierarchies of coupled soliton equations. I, J. Phys. Soc. Jpn., 60, 798-809, (1991) · Zbl 1160.37395
[8] Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27, 1192-1194, (1971) · Zbl 1168.35423
[9] Kakei, S.; Sasa, N.; Satsuma, J., Bilinearization of a generalized derivative nonlinear Schrödinger equation, J. Phys. Soc. Jpn., 64, 1519-1523, (1995) · Zbl 0972.35535
[10] Tian, B.; Gao, Y. T., Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: new transformation with burstons, brightons and symbolic computation, Phys. Lett. A, 359, 241-248, (2006)
[11] Radhakrishnan, R.; Kundu, A.; Lakshmanan, M., Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media, Phys. Rev. E, 60, 3314-3323, (1999)
[12] Agarwal, G. P., Nonlinear fiber optics, (1995), Academic Press New York
[13] Hasegawa, A.; Kodama, Y., Solitons in optical communications, (1995), Oxford University Press Oxford · Zbl 0840.35092
[14] Zhu, S. D., Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method, Chaos Solitons Fract., 34, 1608-1612, (2007) · Zbl 1152.35502
[15] Kim, W. S.; Moon, H. T., Soliton-kink interactions in a generalized nonlinear Schrödinger equation, Phys. Lett. A, 266, 364-369, (2000) · Zbl 0947.35150
[16] Zhang, J.; Dai, C., Bright and dark optical solitons in the nonlinear Schrödinger equation with fourth-order and cubic-quintic nonlinearity, Chin. Opt. Lett., 3, 295-298, (2005)
[17] Shen, Y. J.; Gao, Y. T.; Yu, X.; Meng, G. Q.; Qin, Y., Bell-polynomial approach applied to the seventh-order savada-Kotera-ito equation, Appl. Math. Comput., 227, 502-508, (2014) · Zbl 1364.35317
[18] 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, Nonlinear Dyn., 75, 701-708, (2014) · Zbl 1283.35116
[19] Sun, Z. Y.; Gao, Y. T.; Yu, X.; Liu, Y., Dynamics of bound vector solitons induced by stochastic perturbations: soliton breakup and soliton switching, Phys. Lett. A, 377, 3283-3290, (2013) · Zbl 1302.35351
[20] Sun, Z. Y.; Gao, Y. T.; Liu, Y.; Yu, X., Soliton management for a variable-coefficient modified Korteweg-de Vries equation, Phys. Rev. E, 84, 026606, (2011)
[21] Shen, Y. J.; Gao, Y. T.; Zuo, D. W.; Sun, Y. H.; Feng, Y. J.; Xue, L., Nonautonomous matter waves in a spin-1 Bose-Einstein condensate, Phys. Rev. E, 89, 062915, (2014)
[22] Sun, Z. Y.; Gao, Y. T.; Yu, X.; Liu, Y., Amplification of nonautonomous solitons in the Bose-Einstein condensates and nonlinear optics, Europhys. Lett., 93, 40004, (2011)
[23] Hong, W. P., Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms, Opt. Commun., 194, 217-223, (2001)
[24] Triki, H.; Taha, T. R., Exact analytic solitary wave solutions for the RKL model, Math. Comput. Simul., 80, 849-854, (2009) · Zbl 1186.35210
[25] Zuo, D. W.; Gao, Y. T.; Sun, Y. H.; Feng, Y. J.; Xue, L., Multi-soliton and rogue-wave solutions of the higher-order Hirota system for an erbium-doped nonlinear fiber, Z. Naturforsch. A, 69, 521-531, (2014)
[26] Wu, H. Y.; Fei, J. X.; Zheng, C. L., Self-similar solutions of variable-coefficient cubic-quintic nonlinear Schrödinger equation with an external potential, Commun. Theor. Phys., 54, 55-59, (2010) · Zbl 1213.81127
[27] Angelis, C. D., Self-trapped propagation in the nonlinear cubic-quintic Schrödinger equation: A variational approach, IEEE J. Quant. Electron, 30, 818-821, (1994)
[28] Chin, C.; Kraemer, T.; Mark, M.; Herbig, J.; Waldburger, P.; Ngerl, H.-C.; Grimm, R., Observation of Feshbach-like resonances in collisions between ultracold molecules, Phys. Rev. Lett., 94, 123201-123204, (2005)
[29] Abdullaev, F. K.; Gammal, A.; Tomio, L.; Frederico, T., Stability of trapped Bose-Einstein condensates, Phys. Rev. A, 63, 043604-043614, (2001)
[30] Zhang, W.; Wright, E. M.; Pu, H.; Meystre, P., Fundamental limit for integrated atom optics with Bose-Einstein condensates, Phys. Rev. A, 68, 023605-023610, (2003)
[31] Akhmediev, N.; Ankiewicz, A., Solitons, nonlinear pulses and beams, (1997), Chapman Hall London
[32] Kaplan, A. E., Bistable solitons, Phys. Rev. Lett., 55, 1291-1294, (1985)
[33] Zhou, C.; He, X. T.; Chen, S., Basic dynamic properties of the high-order nonlinear Schrödinger equation, Phys. Rev. A, 46, 2277-2285, (1992)
[34] Afanasjev, V. V.; Chu, P. L.; Kivshar, Yu. S., Breathing spatial solitons in non-Kerr media, Opt. Lett., 22, 1388-1390, (1997)
[35] Artigas, D.; Torner, L.; Torres, J. P.; Akhmediev, N. N., Asymmetrical splitting of higher-order solitons induced by quintic nonlinearity, Opt. Commun., 143, 322-328, (1997)
[36] Kundu, A., Landau-lifschitz and higher order nonlinear systems gauge generated from nonlinear Schrödinger type equations, J. Math. Phys., 25, 3433-3438, (1984)
[37] Calogero, F.; Eckhaus, W., Nonlinear evolution equations, rescalings, model PDES and their integrability: I, Inv. Prob., 3, 229-262, (1987) · Zbl 0645.35087
[38] Levi, D.; Scimiterna, C., The Kundu-eckhaus equation and its discretizations, J. Phys. A, 42, 465203-465210, (2009) · Zbl 1181.35243
[39] Clarkson, P. A.; Tuszynski, J. A., Exact solutions of the multidimensional derivative nonlinear Schrödinger equation for many-body systems of criticality, J. Phys. A, 23, 4269-4288, (1990) · Zbl 0738.35090
[40] Johnson, R. S., On the modulation of water waves in the neighbourhood of kh\(\approx\)1.363, Proc. Roy. Soc. Lond. A, 357, 131-141, (1977) · Zbl 0375.76017
[41] Kodama, Y., Optical solitons in monomode fiber, J. Stat. Phys., 39, 597-614, (1985)
[42] Zhang, J. L.; Wang, M. L.; Li, X. Z., The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation, Phys. Lett. A, 357, 188-195, (2006) · Zbl 1236.81092
[43] Ma, W. X.; Chen, M., Direct search for exact solutions to the nonlinear Schrödinger equation, Appl. Math. Comput., 215, 2835-2842, (2009) · Zbl 1180.65130
[44] Ma, W. X.; Abdeljabbar, A., A bilinear Bäcklund transformation of a (3+1)-dimensional generalized KP equation, Appl. Math. Lett., 25, 1500-1504, (2012) · Zbl 1248.37070
[45] Ma, W. X.; Zhu, Z. N., Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput., 218, 11871-11879, (2012) · Zbl 1280.35122
[46] Ma, W. X.; Zhou, R. G., A coupled AKNS-Kaup-Newell soliton hierarchy, J. Math. Phys., 40, 4419-4428, (1999) · Zbl 0947.35118
[47] Ma, W. X., Bilinear equations and resonant solutions characterized by Bell polynomials, Rep. Math. Phys., 72, 41-56, (2013) · Zbl 1396.35054
[48] Malomed, B. A., Bound solitons in coupled nonlinear Schrödinger equation, Phys. Rev. A, 45, R8321-R8323, (1991)
[49] Seong, N. H.; Kim, D. Y., Experimental observation of stable bound solitons in a figure-eight fiber laser, Opt. Lett., 27, 1321-1323, (2002)
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