We investigate the properties of modulational instability in the Salerno equation in quasione

dimension in Bose-Einstein condensate (BEC). We analyze the regions of modulational instability

of nonlinear plane waves and determine the conditions of its existence in BEC

Abdelaziz, Benseghir

Messikh, Azeddin

Umarov, Bakhram

English

The International Islamic University Malaysia Repository

Australian Journal of Basic and Applied Sciences, 5(5): 25-28, 2011ISSN 1991-8178Modulational Instability In Salerno ModelB.A. Umarov, A. Benseghir, and A. MessikhDepartment of Computational and Theoretical Sciences, Faculty of Science, International IslamicUniversity Malaysia, Kuantan, P.O. Box 141, 25710, MalaysiaDepartment of Computer Science, Faculty of ICT, International Islamic University Malaysia, P.O.Box 10, 50728 Kuala Lumpur, MalaysiaAbstract: We investigate the properties of modulational instability in the Salerno equation in quasi-one dimension in Bose-Einstein condensate (BEC). We analyze the regions of modulational instabilityof nonlinear plane waves and determine the conditions of its existence in BEC. Key words: Modulational instability; Salerno model; Bose-Einstein condensate.INTRODUCTIONThe modulational instability is known phenomenon to be fundamental subject of the theory of nonlinearwaves. This phenomenon consists of the instability of nonlinear plane waves against weak long-scalemodulations with wave numbers (frequency) lower than some critical value. It has been predicted by Banjamenand Feir (Benjamin., 1967) for waves in deep water and by Bespalov and Talanov (Bespalov., 1967) forelectro-magnetic waves in nonlinear media with cubic nonlinearity. Later, it was observed in nonlinear optics(Karpman, 1967; Ostrovskii., 1967; Tanuiti, 1968), plasma physics (Gomez-Gardenes., 2006; Hasegawa, 1970)and condensate matter (Bose-Einstein Condensate, long Josephson junction,...) (Nicolin., 2007; Strecker., 2002).In this work, we determine the regions and conditions of existence of modulational instability in Bose-Einstein condensate in periodic potential trap (optical lattices).II. The Model:It is very important to investigate the modulational instability in different models. We restrict ourselvesto the Salerno model in Bose-Einstein condensate in optical lattice in quasi-1D. This model is a combinationof the discrete non-linear Schrodinger (DNLS) equation (Chris Eilbeck.) with cubic nonlinearity and Ablowitz-Ladik (AL) equation[1]. It is given by ( Gomez-Gardenes., 2006)  (1)where φn is the complex field amplitude at  site of the lattice, µ is the nonlinearity of (AL) equation,   is the nonlinearity of (DNLS) equation.III. Modulational Instability of Nonlinear Plane Wave:The nonlinear discrete equation Eq.(1) has the plane wave solution   (2)where φo, q, ω are amplitude, wave number and frequency respectively. Substituting Eq.(2) into Eq.(1), weobtain.  (3)The Eq.(3) has two solution, one is trivial solution for φn = 0 and other is Corresponding Author: B.A. Umarov,Department of Computational and Theoretical Sciences, Faculty of Science,International Islamic University Malaysia, Kuantan, P.O. Box 141, 25710, MalaysiaE-mail: bakhram@iiu.edu.my25Aust. J. Basic & Appl. Sci., 5(5): 25-28, 2011then, we find the relation between frequency and wave number, which  called “ dispersion relation”.  (4)For unstaggered solution , the dispersion becomes  , for the staggered solution  the dispersion becomes    (Hasegawa, 1970),We investigate when the plane wave solution Eq.(2) is stable against small perturbation , we perturb theplane wave solution slightly such that  (5)and examine evolution of the perturbation using linear stability analysis. then substituting Eq.(5) into Eq.(1) and linearizing at .After simplifying, it become ,   (6)where q = 0 or q = π then sin (q) = 0 so the equation (6) becomes simplifying the above equation, we obtain             (7)Considering the modulational in the form below                             then its complex conjugate is                            ,   substituting the terms  above into Eq.(6), we find    (8)So, we separate the last equation into real part and imaginary part    (9)Then, we consider the modulation in the form below  (10)Where Q and  Ω  are the wave number and frequency of the perturbation respectively.Substituting Eq.(10) into Eq.(9), then we get 26Aust. J. Basic & Appl. Sci., 5(5): 25-28, 2011We rewrite this system of equations in matrix form     We have trivial solution if                       where We compute the determinant of matrix, where ,det (A) = 0 we get   then Where det(A) = 0 Then, we derive the gain which has form  (11)Fig. 1: Modulational Instability regions  27Aust. J. Basic & Appl. Sci., 5(5): 25-28, 2011From the gain of equation (11), we determine the regions of modulational instability. If the expression underroot square is negative then the frequency of perturbation is real, and the modulational is stable. Hence theperturbation leads to small oscillations. If the expression under root square is positive so is positive, thenmodulational is unstable, it means that the modulational is growth exponentially when evolution over time.From the gain of equation, we can see that the unstaggered solution q = 0 is unstable whenever and the staggered solution q = π  is unstable whenever                     in general case where q can take any value, We can distinguish two regions of modulational instability as shown in figure.The one region is fully unstable and other is conditionally stable, it mean that it depends on the wave numberof carrier waves and wave number of perturbation  as shown in the figure.IV. Conclusion:To conclude, we have investigated the modulational instability in a one-dimension described by Salernoequation. The model can applied to BECs in deep optical lattice. Analytical expression for the MI gain spectra isobtained, the regions and conditions of instability of plane wave solutions in the parameter space of the governingSalerno model is determined.ACKNOWLEDGMENTS   This work has been supported by the ministry of Higher Education, Malaysia under research Grant No.IIUM/504/RES/G/14/11/02/FRGS0106-29. REFERENCES Ablowitz M.J, Prinari and A.D. Trubatch, 2004. Discrete and Continuous Nonlinear Schrodinger System   (Cambridge University Press, Cambridge. Bespalov, V. I., V.I. Talanov., 1967. Pisma JETP , 3 471 [ JETP letters, 3, 307 ].Benjamin. T.B., J E. Feir., 1967. Fluid .Mech., 27: 417.Chris Eilbeck. J., Magnus Johanson, The Discrete Nonlinear Schrödinger Equations – 20 Years on, Gomez-Gardenes. J., B.A. Malomed,. L.M. Floria, A.R. Bishop., 2006. Phys. Rev. E 73: 036608.Hasegawa, A., 1970. Phys. Lett. 24: 1165.Karpman V.I., 1967. JETP Lett. 6: 227-143.Nicolin. A.I., 2007. Carretero-Gonzalez. R, Kevrekidis. P. G, Phys. Rev. A 76: 063609.Ostrovskii. L.A., 1967. Sov. Phys. JETP 24: 797..Strecker., K.E., G.B. Partridge., 2002. Truscott. A. G, Hulet. R. G, Nature, 417- 150.Tanuiti , T., H. Washimi, 1968. Phy. Rev. Lett. 21- 209.28

Modulational instability in salerno model

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Modulational instability in salerno model

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