Solution and stability analysis of coupled nonlinear Schrodinger equations

Subject Areas : Applied Mathematics

M. Shahmansouri ¹ , B. Farrokhi ²

1 - Physics Department, Islamic Azad University, Arak Branch, Arak P.O. Box 38135-567, Iran.
2 - Physics Department, Arak University, Arak P.O. Box 38156-879, Iran.

Received: 2011-11-02 Accepted : 2012-02-17 Published : 2014-05-01

Keywords: Stability, Nonlinear Equations, Nonlinear Schrodinger equation,

Abstract :

We consider a new type of integrable coupled nonlinear Schrodinger (CNLS)equations proposed by our self [submitted to Phys. Plasmas (2011)]. The explicitform of soliton solutions are derived using the Hirota's bilinear method.We show that the parameters in the CNLS equations only determine the regionsfor the existence of bright and dark soliton solutions. Finally, throughthe linear stability analysis, the modulational instability condition is given.

References:

[1] B. Farokhi, M. Shahmansouri, and I. Kourakis, Modulated transverse o-
plane dus-lattice wave packets in hexagonal two-dimensional dusty plasma
crystals,
[2] V. I. Karpman, Nonlinear Waves in Dispersive Media, Pergamon, Oxford
(1975).
[3] C. Sulem and P. L. Sulem, The Nonlinear Schrodinger Equation: Self-
Focusing and Wave Collapse, Springer,Berlin (1999).
[4] M. Marklund and P. K. Shukla, Nonlinear collective eects in photonphoton
and photon-plasma interactions,
[5] J. M. Dixon, J. A. Tuszynski and P. J. Clarkson, From Nonlinearity
to Coherence. Universal Features of Nonlinear Behavior in Many-Body
Physics", Cambridge university, Cambridge (1997).
[6] Y. R. Shen, Principles of Nonlinear Optics, Wiley, New York (1984).
[7] A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Oxford
University, New York (1995).
[8] C. X. Miao and G. X. Xu, Global solutions of the Klein-Gordon-Schr?dinger
system with rough data in R2+1,
[9] Xiao-yan Tang and P. K. Shukla, Modulational instability and exact
solution of the nonlinear Schrodinger equation coupled with the nonlinear
Klein-Gordon equation
[10] S. V. Manakov,Zh. Eksp, On the theory of two-dimensional stationary self
focussing of electromagnetic waves,
[11] S. V. Manakov,Zh. Eksp, Analytical solution of the coupled nonlinear
Schrodinger equations,
[12] R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, Inelastic collision
and switching of coupled bright solitons in optical bers,
[13] T. Tsuchida and M. Wadati, The Coupled Modied Korteweg-de Vries
Equations,
[14] T. Tsuchida, N-Soliton Collision in the Manakov Model,

[15] A. V. Mikhailov, The reduction problem and the inverse scattering
method,
[16] V. E. Zakharov and E. I. Schulman, To the integrability of the system of
two coupled nonlinear Schrodinger equations
[17] V. S. Gerdjikov, in Proceedings of the Sixth International Conference on
Geometry, Integrability and Quantization, Varna, Bulgaria, 3-10 June 2004,
edited by I. M. Mladenov and A. C. Hirshfeld, Soa, Softex (2005).
[18] Q. H. Park, and H. J. Shin, Painlev analysis of the coupled nonlinear
Schrodinger equation for polarized optical waves in an isotropic medium
[19] D. N. Christodoulides, and R. I. Joseph, Vector solitons in birefringent
nonlinear dispersive media
[20] C.R. Menyuk, Pulse propagation in an elliptically birefringent Kerr
medium.
[21] V. V. Afanasjev, Yu. S. Kivshar, V. V. Konotop, V. N. Serkin, Dynamics
of coupled dark and bright optical solitons
[22] B. J. Hong, C. C. Yang, L. Wang, Generating dark solitons through crossphase
modulation in optical bers
[23] Yu. S. Kivshar, Stable vector solitons composed of bright and dark pulses
[24] S. Trillo, S. Wabnitz,E. M. Wright, G. Stegeman, Optical solitary waves
induced by cross-phase modulation
[25] Yu. S. Kivshar, S. K. Turitsyn, Optical double layers
[26] A. P. Sheppard, Yu. S. Kivshar, Polarized dark solitons in isotropic Kerr
media
[27] Deng-Shan Wang, Da-Jun Zhang, and Jianke Yang, Integrable properties
of the general coupled nonlinear Schrodinger equations,
[28] H. Q. Zhang, T. Xu, J. Li, B. Tian, Integrability of an N-coupled nonlinear
Schrodinger system for polarized optical waves in an isotropic medium via
symbolic computation,
[29] T. Xu, B. Tian, L.L. Li, X. Lu, C. Zhang, Dynamics of Alfvn solitons in
inhomogeneous plasmas,

[30] R. Hirota, The Direct Method in Soliton Theory, Cambridge University,
Cambridge (2004).
[31] W. Hereman, W. Zhuang, Symbolic Computation of Conserved Densities
for Systems of Nonlinear Evolution Equations,
[32] E. M. E. Zayed, Khaled A. Gepreel, A generalized (G'/G) expansion
method for nding traveling wave solutions of coupled nonlinear evolution
equations,