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$$N$$-soliton solutions and asymptotic analysis for a Kadomtsev-Petviashvili-Schrödinger system for water waves. (English) Zbl 1327.35338
Summary: Under investigation in this paper is a Kadomtsev-Petviashvili-Schrödinger system, which describes the long waves in shallow water. Integrability study has been made through the Painlevé test. Via the Hirota method, the bilinear form and $$N$$-soliton solutions are obtained with an auxiliary variable. Collision of two solitons is found to be elastic by means of the asymptotic analysis. From the graphical descriptions of the two- and three-soliton solutions, it is found that both the bright and dark solitons collide with one another without any change in the physical quantities except for some small phase shifts during the process of each collision.

##### MSC:
 35Q51 Soliton equations 68W30 Symbolic computation and algebraic computation
SYMMGRP
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##### References:
 [1] Osborne, A.R., The inverse scattering transform: tools for the nonlinear Fourier analysis and filtering of Ocean surface waves, Chaos Solitons Fractals, 5, 2623-2637, (1995) · Zbl 1080.86502 [2] Das, G.C.; Sarma, J., Response to comment on a new mathematical approach for finding the solitary waves in dusty plasma, Phys. Plasmas, 6, 4394-4397, (1999) [3] Hong, H.S.; Lee, H.J., Korteweg-de Vries equation of ion acoustic surface waves, Phys. Plasmas, 6, 3422-3424, (1999) [4] Selwyn, G.S.; Heidenreich, J.E.; Haller, K.L., Particle trapping phenomena in radio frequency plasmas, Appl. Phys. Lett., 57, 1876-1878, (1990) [5] Abbasbandy, S.; Zakaria, F.S., Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear Dyn., 51, 83-87, (2008) · Zbl 1170.76317 [6] Bridges, T.J.; Derks, G.; Gottwald, G., Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework, Phys. D, 172, 190-216, (2002) · Zbl 1047.37053 [7] Kaya, D., An explicit and numerical solutions of some fifth-order KdV equation by decomposition method, Appl. Math. Comput., 144, 353-363, (2003) · Zbl 1024.65096 [8] 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) [9] 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 [10] Barnett, M.P.; Capitani, J.F.; Gathen, J.; Gerhard, J., Symbolic calculation in chemistry: selected examples, Int. J. Quantum Chem., 100, 80-104, (2004) [11] Wazwaz, A.M., Exact solutions with solitons and periodic structures for the Zakharov-Kuznetsov (ZK) equation and its modified form, Commun. Nonlinear Sci. Numer. Simul., 10, 597-606, (2005) · Zbl 1070.35075 [12] Teramond, G.F.; Brodsky, S.J., Light-front holography: a first approximation to QCD, Phys. Rev. Lett., 102, 081601-081604, (2009) [13] Schrödinger, E., An undulatory theory of the mechanics of atoms and molecules, Phys. Rev., 28, 1049-1070, (1926) · JFM 52.0965.07 [14] Schrödinger, E., Der stetige übergang von der mikro-zur makromechanik, Naturwissenschaften, 14, 664-666, (1926) · JFM 52.0967.01 [15] Griffiths D.J.: Introduction to Quantum Mechanics. 2nd edn. Pearson Education, Boston (2004) [16] Shanker R.: Principles of Quantum Mechanics. 2nd edn. Plenum Press, New York (1994) [17] Ablowitz, M.J.; Villarroel, J., Solutions to the time dependent Schrödinger and the Kadomtsev-Petviashvili equations, Phys. Rev. Lett., 78, 570-573, (1997) · Zbl 0944.81013 [18] Ablowitz, M.J.; Chakravarty, S.; Trubatch, A.D.; Villarroel, J., A novel class of solutions of the non-stationary Schrödinger and the Kadomtsev-Petviashvili I equations, Phys. Lett. A, 267, 132-146, (2000) · Zbl 0947.35129 [19] Villarroel, J.; Ablowitz, M.J., On the discrete spectrum of the nonstationary Schrödinger equation and multipole lumps of the Kadomtsev-Petviashvili I equation, Commun. Math. Phys., 207, 1-42, (1999) · Zbl 0947.35145 [20] Zhou, X., Inverse scattering transform for the time dependent Schrödinger equation with applications to the KPI aquation, Commun. Math. Phys., 128, 551-564, (1990) · Zbl 0702.35241 [21] Bekir A.: Painlevé test for some (2+1)-dimensional nonlinear equations. Chaos Solitons Fractals 32, 449-445 (2007) · Zbl 1139.37306 [22] Rida, S.Z.; Khalfallah, M., New periodic wave and soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul., 15, 2818-2827, (2010) · Zbl 1222.35162 [23] Elboree, M.K., New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, Appl. Math. Comput., 218, 5966-5973, (2012) · Zbl 1252.35226 [24] Kücükarslan, S., Homotopy perturbation method for coupled Schrödinger-KdV equation, Nonlinear Anal., 10, 2264-2271, (2009) · Zbl 1163.35486 [25] Estévez, P.G., A nonisospectral problem in (2+1) dimensions derived from KP, Inverse Probl., 17, 1043-1052, (2001) · Zbl 0987.35142 [26] Matveev V.B., Salle M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991) · Zbl 0744.35045 [27] Gao, Y.T.; Tian, B., On the non-planar dust-ion-acoustic waves in cosmic dusty plasmas with transverse perturbations, Europhys. Lett., 77, 15001-15006, (2007) [28] Hirota R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004) · Zbl 1099.35111 [29] Freeman, N.C.; Nimmo, J.J., Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique, Phys. Lett. A, 95, 1-13, (1983) · Zbl 0588.35077 [30] Matveev, V.B., Generalized Wronskian formula for solutions of the KdV equations: first applications, Phys. Lett. A, 166, 205-208, (1992) [31] 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) [32] Gao, Y.T.; Tian, B., (3+1)-dimensional generalized Johnson model for cosmic dust-ion-acoustic nebulons with symbolic computation, Phys. Plasmas Lett., 13, 120703-120706, (2006) [33] Feng, Y.J.; Gao, Y.T.; Sun, Z.Y.; Zuo, D.W.; Shen, Y.J.; Sun, Y.H.; Xue, L.; Yu, X., Anti-dark solitons for a variable-coefficient higher-order nonlinear schrodinger equation in an inhomogeneous optical fiber, Phys. Scr., 90, 045201, (2015) [34] 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 [35] Zuo, D.W.; Gao, Y.T.; Xue, L.; Feng, Y.J.; Sun, Y.H., Rogue waves for the generalized nonlinear schrodinger-Maxwell-Bloch system in optical-fiber communication, Appl. Math. Lett., 40, 78-83, (2015) · Zbl 1315.35061 [36] Zuo, D.W.; Gao, Y.T.; Xue, L.; Sun, Y.H.; Feng, Y.J., Rogue-wave interaction for the Heisenberg ferromagnetism system, Phys. Scr., 90, 035201, (2015) [37] Estévez, P.G.; Gordoa, P.R., Darboux transformations via Painlevé analysis, Inverse Probl., 13, 939-957, (1997) · Zbl 0881.35103 [38] Weiss, J.; Tabor, M.; Carnevale, G., The Painlevé property for partial differential equations, J. Math. Phys., 24, 522-526, (1983) · Zbl 0514.35083 [39] Weiss, J., The sine-Gordon equation: complete and partial integrability, J. Math. Phys., 25, 2226-2235, (1984) · Zbl 0557.35107 [40] Fordy, A.P.; Pickering, A., Analysing negative resonances in the Painlevé test, Phys. Lett. A, 160, 347-354, (1991) [41] Conte, R.; Fordy, A.P.; Pickering, A., A perturbative Painlevé approach to nonlinear differential equations, Phys. D, 69, 33-38, (1993) · Zbl 0794.34011 [42] Hereman, W.; Zhuang, W., Symbolic software for soliton theory, Acta Appl. Math., 39, 361-378, (1995) · Zbl 0838.35111 [43] 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
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