Exact Solutions of Discrete Complex Cubic Ginzburg Landau Equation and Their Linear Stability

Transcription

1 Commun. Theor. Phys ) Vol. 56, No. 6, December 15, 2011 Exact Solutions of Discrete Complex Cubic Ginzburg Landau Equation and Their Linear Stability ZHANG Jin-Liang ) and LIU Zhi-Guo Á) School of Mathematics and Statistics, Henan University of Science and Techonology, Luoyang China Received July 13, 2011; revised manuscript received September 14, 2011) Abstract The discrete complex cubic Ginzburg Landau equation is an important model to describe a number of physical systems such as Taylor and frustrated vortices in hydrodynamics and semiconductor laser arrays in optics. In this paper, the exact solutions of the discrete complex cubic Ginzburg Landau equation are derived using homogeneous balance principle and the G /G-expansion method, and the linear stability of exact solutions is discussed. PACS numbers: Dp, Tg, Jr, Yv Key words: discrete complex cubic Ginzburg Landau equation, homogeneous balance principle, G /Gexpansion method, exact solution, linear stability 1 Introduction Discrete solitons in nonlinear lattices have been the focus of considerable attention in diverse branches of science. [1] Discrete solitons are possible in several physical settings, such as biological systems, [2] atomic chains, [3 4] solid state physics, [5] electrical lattices, [6] Bose Einstein condensates [7], photonic structures, [8 13] and nonlinear photonic crystal structure. [14] Thus the discussion on the discrete solitons has been hot topic in the mathematical physics. Thus many researchers have paid attention to the discrete soliton and the nonlinear differentialdifference equation and great progress has been made. In Refs. [15 16], the soliton solutions and Bäcklund transformation of the Toeplitz lattice, the Kupershmidt Five- Field Lattice are presented using the Hirota method. Kou modified the bilinear Bäcklund transformation for the discrete sine-gordon equation and derive variety of solutions by freely choosing parameters from the modified Bäcklund transformation. [17] Wang [18] applied the tanh-function method to study the exact solutions of the Hybrid equation, discrete sine-gordon equation, Toda equation, Ablowitz Ladik equation. The abundant exact solutions for discrete complex cubic Ginzburg Landau equation, [19] discrete complex cubic-quintic Ginzburg Landau equation [20] are derived using the tanh-function method. The modified adomian decomposition method was used to calculate the exact and numerical solutions for discrete complex Ginzburg Landau equation with initial condition. [21] Discrete Ginzburg Landau DGL) models have also been considered in the literature. [10 12] These DGL lattices are quite often used to describe a number of physical systems such as Taylor and frustrated vortices in hydrodynamics [10] and semiconductor laser arrays in optics. [11] Ravoux et al. [13] studied the discrete analog of the complex cubic Ginzburg Landau equation having pattern formation phenomena in mind. In particular, they studied plane wave instability in such systems. Soto Crespo et al. [14] studied the discrete complex cubic Ginzburg Landau DCCGL) equation having several exact solutions. We consider a discrete equation set a model of a dissipative system), viz., the following discrete complex cubic Ginzburg Landau DCCGL) equation [22] i dψ n dt + D 2 iβ ) ψ n+1 2ψ n + ψ n 1 ) + 1 iε) ψ n 2 ψ n+1 + ψ n 1 ) iγψ n = 0, 1) where ψ n is complex variable defined for all integer values of the site index n. ψ n+1 2ψ n + ψ n 1 ) plainly approximates a second derivative term for a continuous system, and thus D is the coefficient of the diffraction term. γ, ε, β are the linear dissipation, cubic nonlinear amplification and filter coefficients, respectively. In the limit of γ = ε = β = 0, Eq. 1) is reduced to the integrable discrete nonlinear Schrödinger equation AL model). [23] The continuous limit of Eq. 1) is the complex Ginzburg Landau equation CGLE). [24] i dψ dt + D 2 iβ ) ψ xx + 1 iε) ψ 2 ψ iγψ = 0. 2) In Ref. [22], the exact solutions of Eq. 1) are derived by using Hirota method, periodic solutions in terms of elliptic Jacobi functions, bright and dark soliton solutions, and constant magnitude solutions with phase shifts are obtained, and in the conclusion of Ref. [22], the authors pointed out that it is very likely that the exact solutions Supported in part by the Basic Science and the Front Technology Research Foundation of Henan Province of China under Grant No and the Doctoral Scientific Research Foundation of Henan University of Science and Technology under Grant No Corresponding author, zhangjin6602@163.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 1112 Communications in Theoretical Physics Vol. 56 of the cubic discrete CGLE are unstable, as was the case with the continuous cubic CGLE. In Ref. [25], the soliton solutions of discrete complex Ginzburg Landau dcgl) equation are numerically studied. The stability, translational invariance and motion properties of dcgle are discussed. Dai [26] applied the extended tanh-expansion method to obtain the exact solutions of Eq. 1) including the bright soliton, dark soliton and alternating phase soliton, the stability analysis of the exact solutions of Eq. 1) was not presented. G /G-expansion method was firstly introduced in Ref. [27] and has attracted a wide interest. This method has been used to explore the exact solutions of some nonlinear evolution equations, such as: mkdv equation with variable coefficients, [28] the 2+1)-dimensional Nizhnik Novikov Vesselov NNV) equation, the 2+1)- dimensional Broer Kaup BK) equation the 2+1)- dimensional Kadomstev Petviashvili KP), [29] CKdVmKdV, the reation-diffusion equation, the compound KdV-Burgers equation and the generalized shallow water wave equation, [30] Fisher equation and CKdV, [31] 3+1) potential YTSF, [32] 2+1)-dimensional CGKP, KdV, and 2+1)-dimensional Burgers equations with variable coefficients, [33] etc., and some nonlinear differentialdifference equations, such as: 2+1)-dimensional Toda lattice, [34] the Toda lattice, discrete nonlinear Schrödinger equation, [35] the Ablowitz Ladik lattice system, [36] etc. Some comments on the G /G-expansion method are made. [37 39] From our application and investigation into the G /G-expansion method, it is to see that the relation between the exact solutions of nonlinear ODE with solutions of second linear sub-ode can be established using the G /G-expansion method The other subsidiary equation method can establish the relation between the exact solutions of nonlinear ODE with solutions of nonlinear sub-ode, the sub-odes are Riccati equation, Jacobi elliptic equation, etc., [40] these sub-odes are nonlinear), another advantage of the G /G-expansion method is to see that the exact solutions of nonlinear ODE have more free constants. In this paper, Eq. 1) is studied and the hyperbolic function solitary wave solutions, trigonometric function periodic wave solutions and rational wave solutions with the arbitrary parameters are obtained using the extended G /G-expansion method. [41] By using the perturbation method, the linear stability of exact solutions obtained in this paper is analyzed. 2 Exact Solutions of Discrete Complex Cubic Ginzburg Landau DCCGL) Equation Since ψ n is complex, we suppose where ψ n n, t) = e iθn U n ξ n ), 3) θ n = d 1 n + c 1 t + ζ 1, ξ n = d 2 n + c 2 t + ζ 2, 4) ζ 1, ζ 2 are arbitrary constants, d 1, c 1, d 2, and c 2 are constants determined later. Substituting Eq. 3) into Eq. 1) and setting the real and imaginary parts of the resulting equations to zero yields D ) cosd 1 ) 2 + U2 n U n+1 + U n 1 ) + sind 1 )β + εun) 2 U n+1 U n 1 ) c 1 + D)U n = 0, 5) c 2 U n cosd 1 )β + εun)u 2 n+1 + U n 1 ) D ) + sind 1 ) 2 + U2 n U n+1 U n 1 ) + γ)u n = 0. 6) According to the homogeneous balance principle, [42 45] we suppose G ξ n ) ) [ G ξ n ) ) 2 / ] U n ξ n ) = a 0 +a 1 + b 1 σ 1 + µ, Gξ n ) Gξ n ) σ = ±1, a 1 b 1 = 0, a b2 1 0, 7) where Gξ n ) satisfies Case 1 µ < 0, b 1 = 0 If µ < 0, b 1 = 0, we have G ξ n ) + µgξ n ) = 0. 8) [ G ξ n ) U n±1 = a 0 + a 1 Gξ n ) ± µ tanh ]/[ µd 2 ) 1 ± 1 G ξ n ) ] µ Gξ n ) tanh µd 2 ). 9) Substituting Eqs. 7) and 9) into Eqs. 5) 6) and using Eq. 8), the exact solutions of Eq. 1) are derived as: ψ n t) = γ A 1 sinh µξ n + A 2 cosh µξ n 2ε A 1 cosh µξ n + A 2 sinh exp i γt ) µξ n ε + iζ 1, 10) where ξ n = ±n/ µ)arctanh γ/ + ζ 2, ζ 1, ζ 2 arbitrary constants, γ/ε < 0, 0 < γ/β < 2. γ 4β A 1 sinh µξ n + A 2 cosh µ ξ n ψ n t) = 2ε A 1 cosh µ ξ n + A 2 sinh exp i γt ) µ ξ n ε + iζ 1 + inπ, 11) where ξ n = ±n/ µ)arctanh 4β γ)/ + ζ 2, ζ 1, ζ 2 arbitrary constants, γ 4β/ε > 0, 2 < γ/β < 4. When A 1 0, A 2 2 < A2 1, discrete solitons of Eq. 1) can be derived as Case 1.1 Dark soliton ψnt) 1 = γ2ε [ tanh γ ) ] ±n arctanh + ζ 2 exp i γt ) ε + iζ 1, 12)

3 No. 6 Communications in Theoretical Physics 1113 where ζ 1, ζ 2 arbitrary constants, γ/ε < 0, 0 < γ/β < 2. Case 1.2 Alternating phase dark soliton [ γ 4β ψnt) 2 4β γ ) ] = tanh ±n arctanh + ζ 2 exp i γt ) 2ε ε + iζ 1 + inπ, 13) where ζ 1, ζ 2 arbitrary constants, γ 4β/ε > 0, 2 < γ/β < 4. Fig. 1 a) Dark soliton 12) when γ = 0.01, ε = , β = 0.04, ζ 1 = ζ 2 = 0, t = 0; b) Alternating phase dark soliton 13) when γ = 0.03, ε = , β = 0.01, ζ 1 = ζ 2 = 0, t = 0. Remark 1 Here we only present the soliton solutions of Eq. 1), the other type solutions are omitted for simplicity. Case 2 µ < 0, a 1 = 0 Similar to Case 1, the exact solutions of Eq. 1) are obtained as Case 2.1 Bright soliton ψnt) 3 γ = 2 4βγ [ sech ±n arccosh 1 γ ) ] + ζ 2 exp i γt ) 4βε ε + iζ 1, 14) where ζ 1, ζ 2 arbitrary constants, γ/β < 0, γ/ε < 0. Case 2.2 Alternating phase bright soliton ψn 4 t) = γ 2 4βγ [ γ ) sech ±n arccosh 4βε 1 + ζ 2 ]exp i γt ) ε + iζ 1 + inπ, 15) where ζ 1, ζ 2 arbitrary constants, γ/β > 4, γ/ε > 0. Fig. 2 a) Bright soliton 14) when γ = 0.01, ε = , β = 0.04, ζ 1 = ζ 2 = 0, t = 0; b) Alternating phase bright soliton 15) when γ = 0.01, ε = , β = 0.001, ζ 1 = ζ 2 = 0, t = 0.

4 1114 Communications in Theoretical Physics Vol. 56 Case 3 µ > 0, b 1 = 0 Similar to Case 1, the exact solutions of Eq. 1) are obtained as γ A 1 cos µ ξ n A 2 sin µ ξ n ψ n t) = 2ε A 1 sin µ ξ n + A 2 cos exp i γt ) µ ξ n ε + iζ 1, 16) where ζ 1, ζ 2 arbitrary constants, γ/β < 0, γ/ε > 0. 4β γ ψ n t) = 2ε ξ n = ± n arctan γ ) + ζ 2, µ A 1 cos µ ξ n A 2 sin µ ξ n A 1 sin µξ n + A 2 cos µ ξ n exp i γt ) ε + iζ 1 + inπ, 17) where ξ n = ±n/ µ) arctan γ/ 2 + ζ 2, ζ 1, ζ 2 arbitrary constants, γ/β > 4, 4β γ)/ε > 0. When A 2 0, A 2 1 < A2 2, the discrete periodical wave solutions of Eq. 1) are obtained as Case 3.1 Oscillatory tan solution. γ ψn [±n 5 t) = 2ε tan arctan γ ) ] + ζ 2 exp i γt ) ε + iζ 1, 18) where ζ 1, ζ 2 arbitrary constants, γ/β < 0, γ/ε > 0. Case 3.2 Alternating phase oscillatory tan solution [ tan ±n arctan ψ 6 n t) = 4β γ 2ε γ 2 ) + ζ 2 ]exp i γt ) ε + iζ 1 + inπ, 19) where ζ 1, ζ 2 arbitrary constants, γ/β > 4, 4β γ)/ε > 0. Remark 2 In Refs. [22] and [19], value ranges of β, γ, ε are γ/β > 4, 4β γ)/ε > 0 in solution 19), but solution 19) is only verified by Mathematica 5.0 if value ranges of β, γ, ε are γ/β = 5, 4β γ)/ε > 0 in this paper. Fig. 3 a) Oscillatory tan solution 18) when γ = 0.01, ε = , β = 0.04, ζ 1 = ζ 2 = 0, t = 0; b) Alternating phase oscillatory tan solution 19) when γ = 0.005, ε = , β = 0.001, ζ 1 = ζ 2 = 0, t = 0. Case 4 µ > 0, a 1 = 0 Similar to Case 1, the exact solutions of Eq. 1) are obtained as Case 4.1 Oscillatory sec solution ψn 7 t) = γ 2 4βγ [ sec ±n arccos 1 γ ) ] + ζ 2 exp i γt ) 4βε ε + iζ 1, 20) where ζ 1, ζ 2 arbitrary constants, 0 < γ/β < 4, γ/ε < 0. Case 4.2 Alternating phase oscillatory sec solution ψnt) 8 γ = 2 4βγ [ γ ) sec ±n arccos 4βε 1 + ζ 2 ]exp i γt ) ε + iζ 1 + inπ, 21)

5 No. 6 Communications in Theoretical Physics 1115 where ζ 1, ζ 2 arbitrary constants, 0 < γ/β < 4, γ/ε < 0. Fig. 4 a) Oscillatory sec solution 20) when γ = 0.01, ε = , β = 0.04, ζ 1 = ζ 2 = 0, t = 0; b) Alternating phase oscillatory sec solution 21) when γ = 0.01, ε = , β = 0.04, ζ 1 = ζ 2 = 0, t = 0. 3 Stability Analysis In order to study the linear stability of ψ j n j = 1, 2,..., 8), we suppose ψ n t) = ψ j n + δψ nt)e iωt, 22) where δψ n t) is a small perturbation. Substituting Eq. 22) into Eq. 1) and retaining only terms linear in the perturbation yields iδ ψ D ) n + ω iγ)δψ n + 2 iβ δψ n+1 2δψ n + δψ n 1 ) + 1 iε)[ψ j n+1 + ψj n 1 )ψ j n δψ n + δψ n ) + δψ n+1 + δψ n 1 ) ψ j n 2 ] = 0. 23) By splitting the perturbation δψ n t) into real parts δu n and imaginary parts δv n, δψ n = δu n +iδv n and introducing the two real vectors δu = {δu n }, δv = {δu n }, and the two real matrices A = {A nm } and B = {B nm } β [ A nm = δ nm+1 + δ nm 1 ) ε + ψj n 2) + δ nm + 2ψ j n+1 ε + ψj n 1 ) + γ ε β B nm = δ nm+1 + δ nm 1 ) ε + ψj n 2) [ + δ nm ε + γ ε where m ± 1 in the Kronecker δ means: m ± 1 mod N. Then Eq. 23) can be written compactly as ], 24) ], 25) δ V + AδU + εbδv = 0, δ U + ε)aδu + BδV = 0. 26) When ε is very small ε < ), Eq. 26) are approximated by From Eq. 27), it is easy to obtain that δ V + AδU = 0, δ U + BδV = 0. 27) δ V + ABδV = 0, δü + BAδU = 0. 28) The eigenvalue spectrum of the matrices AB and BA determines the stability of the exact solutions. If it contains negative or complex) eigenvalues, the solution is unstable. [46 47] It is easy to obtain that the solutions 13) 15), 18) 21) are unstable because it contains negative eigenvalues from Figs. 5b) 5h). From Fig. 6 the enlarged graph of Fig. 5a)), we can obtain that AB of solutions 12) has negative eigenvalue, thus solution 12) is unstable.

6 1116 Communications in Theoretical Physics Vol. 56 Fig. 5 a) h) are the distribution graphs of the minimal eigenvalues of AB from 4 order to 31 order) of the exact solutions shown by Figs Fig. 6 Enlarged graph of Fig. 5a). 4 Conclusions and Discussion i) In Sec. 2, the hyperbolic function solitary wave solutions, trigonometric function periodic wave solutions and rational wave solutions with the arbitrary parameters to Eq. 1) are obtained using extended G /G-expansion method. The exact solutions 13) and 18) in Ref. [22] and the exact solutions 29) 30) of Eq. 1) in Ref. [19] are respectively derived from solutions 10) 11) in this paper when A 1 0, A 2 2 < A2 1. The exact solutions 15) and 20) in Ref. [22] and the exact solutions 31) 32) of Eq. 2) in Ref. [19] are respectively derived from Eqs. 16) 17) in this paper when A 2 0, A 2 1 < A2 2. From comparing the representations of exact solutions using extended G /G-expansion method [40] with ones of exact solutions using G /G-expansion method, [27 33] it is easy to obtain that the representations of exact solutions using extended G /G-expansion method have more items n G ) i 1 [ G ) 2 / {b i σ 1 + µ] }, G G i=1 thus the exact solutions of Eq. 1) derived by extended G /G-expansion method in this paper are more general than ones presented using G /G-expansion method. ii) The linear stability of the bright solitons, dark solitons, alternating phase bright solitons, alternating phase dark solitons and trigonometric function periodic wave so-

7 No. 6 Communications in Theoretical Physics 1117 lutions are analyzed, and the exact solutions obtained in Sec. 2 are unstable. iii) The analysis of the minimal eigenvalues of the perturbation equation shows that minimal eigenvalue λ tends to zero gradually as cubic nonlinear amplification ε a small value) enlarges. Fig. 7 The illustration of changing cases of the minimal eigenvalue λ of each order dark soliton 12) with γ = 0.01, β = 0.04) as ε enlarges gradually. Figure 7 shows that amplitude of vibration to the perturbation equations decrease as cubic nonlinear amplification ε a small value) enlarges gradually. Thus the cubic nonlinear amplification ε has remarkable influence on the stability of the exact solutions. Now we have a problem which could the minimal eigenvalue λ be zero or greater than zero if ε is set to appropriate value or could the exact solutions be stable via changing the value of ε? From the discussions of stability analysis in Ref. [25], it is impossible. Thus the cubic nonlinear amplification ε is essential for the stability of these solutions. The numerical simulation Refs. [25,48 54]) is one of the important methods in the investigation into the stability properties, the dynamics and other properties of the discrete) soliton for the nonlinear Schrödinger-type equations. We have applied the numerical simulation method to study the properties of the discrete soliton for the discrete complex cubic Ginzburg Landau DCCGL) equation 1), and obtained some valuable results. These results will be presented in another paper recently. Acknowledgments The authors would like to express their sincere thanks to the referee for the valuable suggestions. References [1] P.G. Kevrekidis, K. /O. Rasmussen, and A.R. Bishop, Int. J. Mod. Phys. B ) [2] A.C. Scott, Philos. Trans. R. Soc. London A ) 423. [3] A.C. Scott and L. Macneil, Phys. Lett. A ) 87. [4] A.J. Sievers and S. Takeno, Phys. Rev. Lett ) 970. [5] W.P. Su, J.R. Schieffer, and A.J. Heeger, Phys. Rev. Lett ) [6] P. Marquii, J.M. Bilbaut, and M. Remoissenet, Phys. Rev. E ) [7] A. Trombettoni and A. Smerzi, Phys. Rev. Lett ) [8] F. Lederer, S. Darmanyan, and A. Kobyakov, in: Spatial Solitons, eds. S. Trillo and W.E. Toruellas, Springer, Berlin 2001) p [9] D.N. Christodoulides and R.I. Joseph, Opt. Lett ) 794. [10] H.S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J.S. Aitchison, Phys. Rev. Lett ) [11] O. Bang and P. Miller, Opt. Lett ) [12] M. Ablowitz and Z.H. Musslimani, Phys. Rev. Lett ) [13] A.A. Sukhorukov and Y. Kivshar, Phys. Rev. E ) [14] D.N. Christodoulides and N.K. Efremidis, Opt. Lett ) 568. [15] X.B. Hu and W.X. Ma, Phys. Lett. A ) 161. [16] H.W. Tam and X.B. Hu, Appl. Math. Lett ) 987. [17] X. Kou, et al., Commun. Theor. Phys ) 545. [18] Z. Wang, Comput. Phys. Commun ) [19] C.Q. Dai, X. Cen, and S.S. Wu, Comput. Math. Appl ) 55. [20] C.Q. Dai, et al., Opt. Commun ) 309. [21] Y.Y. Wang, et al., Commun. Theor. Phys ) 81. [22] K. Maruno, A. Ankiewicz, and N. Akhmediev, Opt. Commun ) 199. [23] M. Ablowitz and J.F. Ladik, Stud. Appl. Math ) 213. [24] I.S. Aranson and L. Kramer, Rev. Mod. Phys ) 99. [25] J.M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Lett. A ) 126. [26] C.Q. Dai, X. Cen, and S.S. Wu, Comput. Math. Appl ) 55. [27] M.L. Wang, X.Z. Li, and J.L. Zhang, Phys. Lett. A ) 417. [28] S. Zhang, J.L. Tong, and W. Wang, Phys. Lett. A ) [29] J. Zhang, X.L. Wei, and Y.J. Lu, Phys. Lett. A ) [30] E.M.E. Zayed and K.A. Gepreel, J. Math. Phys ) [31] T. Ozis and I. Aslan, Commun. Theor. Phys ) 577.

8 1118 Communications in Theoretical Physics Vol. 56 [32] E.M.E. Zayed, J. Phys. A: Math. Theor ) [33] L.L. Zhang and X.Q. Liu, Commun. Theor. Phys ) 784. [34] S. Zhang, L. Dong, J.M. Ba, and Y.N. Sun, Phys. Lett. A ) 905. [35] I. Aslan, Appl. Math. Comput ) [36] I. Aslan, Appl. Math. Comput ) [37] C.P. Liu, Commun. Theor. Phys ) 985. [38] Y.L. Zhao, Y.P. Liu, and Z.B. Li, Chin. Phys. B ) [39] P. Zhu, Commun. Theor. Phys ) 206. [40] Taogetusang, Acta Phys. Sin ) [41] S.M. Guo and Y.B. Zhou, Appl. Math. Comput ) [42] M.L. Wang, Phys. Lett. A ) 169. [43] M.L. Wang, Phys. Lett. A ) 279. [44] M.L. Wang and Y.M. Wang, Phys. Lett. A ) 211. [45] J.L. Zhang, et al., Chin. Phys ) 245. [46] A. Khare, K. /O. Rasmussen, M.R. Samuelsen, and A. Saxena, J. Phys. A: Math. Gen ) 807. [47] M. Syafwan, H. Susanto, and S.M. Cox, Phys. Rev. E ) [48] J.L. Dai, et al., J. Phys. B: At. Mol. Opt. Phys ) [49] J.L. Chen, et al., Commun. Theor. Phys ) 335. [50] L. Wu, et al., Phys. Rev. A ) [51] C.M. Schober, Phys. Lett. A ) 140. [52] C. Wang, et al., Phys. Rev. E ) [53] D.E. Pelinovsky, P.G. Kevrekidis, and D.J. Frantzeskakis, Physica D ) 1. [54] D.E. Pelinovsky, P.G. Kevrekidis, and D.J. Frantzeskakis, Physica D ) 20.