LIE SYMMETRY ANALYSIS AND EXACT SOLUTIONS TO N-COUPLED NONLINEAR SCHRÖDINGER S EQUATIONS WITH KERR AND PARABOLIC LAW NONLINEARITIES

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1 LIE SYMMETRY ANALYSIS AND EXACT SOLUTIONS TO N-COUPLED NONLINEAR SCHRÖDINGER S EQUATIONS WITH KERR AND PARABOLIC LAW NONLINEARITIES YAKUP YILDIRIM 1, EMRULLAH YAŞAR 1, HOURIA TRIKI 2, QIN ZHOU 3, SEITHUTI P. MOSHOKOA 4, MALIK ZAKA ULLAH 5, ANJAN BISWAS 4,5, MILIVOJ BELIC 6 1 Department of Mathematics, Faculty of Arts and Sciences, Uludag University, Bursa, Turkey 2 Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, Annaba, Algeria 3 School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan , People s Republic of China 4 Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria-0008, South Africa 5 Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah-21589, Saudi Arabia biswas.anjan@gmail.com 6 Science Program, Texas A & M University at Qatar, P.O. Box 23874, Doha, Qatar Received May 26, 2017 Abstract. This paper addresses N-coupled nonlinear Schrödinger s equation with spatio-temporal dispersion for Kerr and parabolic laws of nonlinearity by the aid of Lie symmetry analysis. We systematically construct similarity reductions to the derived ordinary differential equations by Lie group analysis. These equations lead to exact solutions. Key words: Lie symmetry analysis; nonlinear Schrödinger s equation; optical solitons; spatio-temporal dispersion. 1. INTRODUCTION The complex dynamics of optical soliton propagation is governed by the nonlinear Schrödinger s equation NLSE). There are several advances that have been made with this generic nonlinear evolution equation. While this model has been mostly studied in scalar form, the vector-coupled NLSE has also been addressed in the context of birefringent optical fibers. This paper will focus on N-coupled NLSE that appears in the context of parallel propagation of solitons for large data transmission across trans-continental and trans-oceanic distances. There are several integration algorithms that are applied to reveal results for NLSE in polarization-preserving as well as birefringent optical fibers. This paper will apply one of the most powerful mathematical approaches to study N-coupled NLSE. It is the Lie symmetry analysis. This classic mathematical methodology will never be rusty in any area of Romanian Journal of Physics 63, ) v.2.1* #b79a7948

2 Article no. 103 Yakup Yıldırım et al. 2 mathematical physics or other fields of applied mathematics. In the past, several other techniques have been implemented to extract solitons and other solutions to such a system. They are Kudryashov s method, G /G-expansion scheme, method of undetermined coefficients [1] and several others. Performance enhancement in soliton propagation across the globe can only be achieved through the dense wavelength division multiplexing DWDM) technology implementation, which is modeled by N-coupled NLSEs. This paper studies DWDM systems in nonlinear optical fibers with Kerr and parabolic laws of optical nonlinearity. The unique soliton dynamics has been extensively studied in optical fibers, photonic crystal fibers PCFs), metamaterials, and metasurfaces [1-24]. Therefore, it is about time to focus and pay attention to DWDM optical systems. The details of Lie symmetry analysis along with its implementation to such DWDM systems are studied in this paper. 2. THE MODEL The dynamics of optical soliton propagation through a DWDM system is governed by the N-coupled NLSEs. There are two types of nonlinear media and associated optical nonlinearities that will be studied in this paper. These are Kerr law nonlinear media and the parabolic law nonlinear media. For NLSE, in addition to usual group-velocity dispersion GVD), spatiotemporal dispersion STD) is included. STD makes the governing model well-posed as opposed to the presence of GVD alone. Thus, the model is discussed in the following subsections based on the types of nonlinearity in question KERR LAW For Kerr law nonlinearity, the generic DWDM model reads [1] iq l) t + a l q xx l) + b l q l) xt + c l q l) 2 N q + n) α ln 2 ql) = 0, 1) where 1 l N. The first term in 1) on left-hand side is the evolution term, while a l represents the coefficient of GVD. Here b l represents the STD. Then, c l is the coefficient of self-phase modulation SPM) while α ln are the coefficients of crossphase modulation XPM). The independent variables are x and t, which represents the spatial and temporal variables, respectively. The dependent variable is q l) x,t) that gives the soliton profile in every single channel. It must be noted that STD terms are deliberately included for the problem to be well-posed. c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948

3 3 Lie symmetry analysis and exact solutions to N-coupled NLSEs Article no PARABOLIC LAW For parabolic law nonlinearity, DWDM model reads [1] iq l) t + a l q xx l) + b l q l) q xt + l) c l 2 q n) + α ln 2 q l) q l) + d l 4 { + q n) 2 q n) β ln 2 q l) + γ ln 2} q l) = 0 2) for 1 l N. In 2), the SPM terms are the coefficients of c l and d l, while the XPM coefficients are α ln, β ln and γ ln. The remaining parameters have the same definition as in Kerr law nonlinear medium. In mathematical physics, the generic equations 1) and 2) fall under the category of nonlinear evolution equations NLEEs). 3. SYMMETRY ANALYSIS AND SYMMETRY REDUCTIONS The investigation of exact solutions to NLEEs is a quite important task in the nonlinear science. In this regard, many powerful methods have been developed in the last three decades such as inverse scattering method, Darboux method, Hirota bilinear method, ansatz method, multiple-exp function method, simplest equation method etc. [2 5, 7]. Since the end of the 19th century, the symmetry study plays an important role in almost all the scientific fields. Inspired by Galois s researches on algebraic equations, S. Lie showed that considered differential equations can be invariant with respect to continuous transformation groups. Obtaining the symmetry reductions and thereby group invariant solutions are possible by Lie generators corresponding to the transformation groups. For the detailed studies on applications of Lie group analysis to differential equations, we suggest to readers to see Refs. [8 11] KERR LAW a l and b l 0 are arbitrary constants Substituting the complex-valued functions q l) x,t) into the system 1) and then decomposing into real and imaginary parts yields a pair of relations. The real part gives Im q l) t ) +a l Re q l) xx ) +b l Re q l) xt ) + c l q l) 2 N q + n) α ln 2 Re q l)) = 0 c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948 3)

4 Article no. 103 Yakup Yıldırım et al. 4 while the imaginary part gives Re q l) t ) + a l Im q l) xx ) + b l Im q l) xt ) + c l q l) 2 N q + n) α ln 2 Im q l)) = 0 where 1 l N. Let us consider the Lie group of point transformations t = t + ɛτ x,t, Re q l)), Im q l))) + O ɛ 2) x = x + ɛξ x,t, Re q l)) = Re q l)) + ɛη l x,t, Im q l)) = Im q l)) + ɛφ l x,t, Re q l)), Re q l)), Re q l)), Im q l))) + O ɛ 2) Im q l))) + O ɛ 2) Im q l))) + O ɛ 2) with small parameter ɛ 1 and 1 l N. The vector field associated with the above group of transformations can be written as V = ξ x,t, Re q l)), Im q l))) x +τ x,t, Re q l)), + η l x,t, Re q l)), + φ l x,t, Re q l)), Im q l))) t Im Im q l))) q l))) Re q l)) Im q 4) l)) 5) The symmetries of equations 3)-4) will be generated by the vector field of the form 5). Applying the second prolongation pr 2) V of V to equations 3)-4) and splitting on the derivatives of Re q l)), Im q l)) leads to the following overdetermined system of linear partial differential equations: τ x = τ t = τ Re q l)) = τ Im q l)) = 0 c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948

5 5 Lie symmetry analysis and exact solutions to N-coupled NLSEs Article no. 103 ξ x = ξ t = ξ Re q l)) = ξ Im q l)) = 0 φ l x = φ l t = φ N l Im q l)) = φ l Re q n)) = 0 φ l Re q l)) = φ l Re q l)), η l = Im q l) ) φ l Re q l)) where 1 l N. Solving the above equations we obtain the values of ξ, τ, η l, and φ l τ = C 1 ; ξ = C 2 η l = C l+2 Im q l)) φ l = C l+2 Re q l)) 6) where C 1, C 2, and C l+2 are arbitrary constants and 1 l N. As a result we obtain the infinitesimal generators of the corresponding Lie algebra of Eq. 1) are given by V l+2 = Re V 1 = x ; V 2 = t Im q l)) Im q l)) q l)) where q l) are complex-valued functions and 1 l N. Re q l)), a l is an arbitrary nonzero constant and b l = 0 In this case, Eq. 1) becomes iq l) t + a l q xx l) + c l q l) 2 N q + n) α ln 2 ql) = 0 7) The infinitesimal generators of the corresponding Lie algebra of Eq.7) are given by V l+2 = Re V N+3 = 2t t + x x + N V N+4 = t x + N V 1 = x ; V 2 = t q l)) Im q l)) Im q l)) Re q l)) { } Re q l)) Re q l)) Im q l)) Im q l)) { xim q l)) 2a l Re xre q l) ) } q l)) + 2a l Im q l)), c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948

6 Article no. 103 Yakup Yıldırım et al. 6 where q l) are complex-valued functions and 1 l N b l is an arbitrary nonzero constant and a l = 0 In this case, Eq. 1) becomes iq l) t + b l q l) xt + c l q l) 2 N q + n) α ln 2 ql) = 0 8) The infinitesimal generators of the corresponding Lie algebra of Eq. 8) are given by V 1 = x ; V 2 = t V l+2 = Re q l)) Im q l)) Im q l)) Re q l)) V N+3 = t N t + { Re q l)) 2 Re Im q l) ) } q l)) 2 Im q l)) { b lre q l)) 2xIm q l)) 2b l V N+4 = x x + Re q l)) b lim q l) ) + 2xRe q l)) 2b l } Im q l)) where q l) are complex-valued functions and 1 l N. To obtain the symmetry reductions of equations 3)-4), we have to solve the characteristic equation dx ξ = dt τ = dre q l) ) η l = dim q l) ), 9) φ l where ξ, τ, η l, and φ l are given by 6) and 1 l N. To solve 9), we consider the following cases: i) V 1 + kv l+2 ii) V 2 + µv l+2 Case i) V 1 + kv l+2 Solving the characteristic equation 9), we have the following similarity variables ξ = t q l) x,t) = F l ξ)expikx + G l ξ))) 10) where ξ is a new independent variable and F l and G l are new dependent variables and 1 l N. c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948

7 7 Lie symmetry analysis and exact solutions to N-coupled NLSEs Article no. 103 Substituting equations 10) into system 3)-4), we immediately obtain the reduced equations, which after separating imaginary and real parts read kb l + 1)F l = 0 c l Fl 3 kb l F l G l + α ln Fn 2 F l k 2 a l F l F l G l = 0 11) where 1 l N. We obtain the following solution of ordinary differential equations ODEs) 11) F l = C l ) c l Cl 2 a lk 2 + N α ln Fn 2 ξ G l = + κ l 12) b l k + 1 where C l and κ l are arbitrary constants, and 1 l N. The corresponding solution of the system 1) is given by ) c l Cl 2 q l) a lk 2 + N α ln Fn 2 t x,t) = C l exp i kx + + κ l b l k + 1 Case ii) V 2 + µv l+2 Solving the characteristic equation, the similarity variables are ξ = x q l) x,t) = F l ξ)exp{i[µt + G l ξ)]}, 13) where ξ is a new independent variable, and F l and G l are new dependent variables. Using 13) in system 3)-4), we obtain the following system of ODEs a l F l G l + 2a lf l G l + b lµf l = 0 14) c l Fl 3 a l F l G 2 l b l µf l G l + α ln Fn 2 Solving the equations 14), we have F l µf l + a l F l = 0 15) G l = b lµfl 2 κ 2 l 2a l Fl 2, 16) c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948

8 Article no. 103 Yakup Yıldırım et al. 8 where κ l are arbitrary constants and 1 l N. Using 16) in equations 15), we have c l Fl 3 bl µfl 2 κ 2 ) 2 l 4a l Fl 3 + b lµ b l µfl 2 κ 2 ) l + α 2a l F l ln Fn 2 F l µf l + a l F l = 0 17) The corresponding solution of the system 1) is given by { [ q l) x,t) = F l ξ)exp i µt bl µfl 2 κ 2 l 2a l Fl 2 dξ + θ l )]}, 18) where θ l are arbitrary constants and 1 l N. ξ is given by 13) and F l are given by 17). by 3.2. PARABOLIC LAW a l and b l 0 are arbitrary constants The infinitesimal generators of the corresponding Lie algebra of 2) are given V l+2 = Re V 1 = x ; V 2 = t Im q l)) Im q l)) q l)) where q l) are complex-valued functions and 1 l N. Re q a l is an arbitrary nonzero constant and b l = 0 In this case, Eq. 2) becomes iq l) t + a l q xx l) q l) + c l 2 q n) + α ln 2 q l) l)), 19) q l) + d l 4 { + q n) 2 q n) β ln 2 q l) + γ ln 2} q l) = 0 20) The infinitesimal generators of the corresponding Lie algebra of Eq. 20) are given by V 1 = x ; V 2 = t V l+2 = Re q l)) Im q l)) Im q l)) Re q l)) c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948

9 9 Lie symmetry analysis and exact solutions to N-coupled NLSEs Article no. 103 V N+3 = t x + N { xim q l)) 2a l Re q xre q l) ) l)) + 2a l where q l) are complex-valued functions and 1 l N b l is an arbitrary nonzero constant and a l = 0 In this case, Eq. 2) becomes iq l) t + b l q l) q xt + l) c l 2 q n) + α ln 2 q l) Im q l)) q l) + d l 4 { + q n) 2 q n) β ln 2 q l) + γ ln 2} q l) = 0 21) The infinitesimal generators of the corresponding Lie algebra of Eq. 21) are given by V l+2 = Re V N+3 = t t x x + N V 1 = x ; V 2 = t Im q l)) Im q l)) Re q l)) { xim q l)) Re xre q l) ) q l)) + q l)) b l b l }, Im q l)) where q l) are complex-valued functions and 1 l N. The similarity variables for system 2), corresponding to the vector field µ 2 V 1 + µ 1 V 2 + µ 3 V l+2 are given by the vector fields 19) ξ = µ 1 x µ 2 t }, q l) x,t) = F l ξ)exp{i[µ 3 x + G l ξ)]}, 22) where ξ is the new independent variable and F l,g l are the new dependent variables. Using the similarity variables 22) in system 2) and separating the real and imaginary parts, we obtain µ 2 F l + 2a lµ 2 1F l G l b lµ 2 µ 3 F l b lµ 1 µ 2 F l G l + 2a lµ 1 µ 3 F l 2b l µ 1 µ 2 F l G l + a lµ 2 1F l G l = 0, µ 2 F l G l a lµ 2 1F l G 2 l 2a l µ 1 µ 3 F l G l a lµ 2 3F l +a l µ 2 1F l +b lµ 1 µ 2 F l G 2 l +b l µ 2 µ 3 F l G l c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948

10 Article no. 103 Yakup Yıldırım et al. 10 b l µ 1 µ 2 F l + c l F 3 l + α ln Fn 2 F l + γ ln Fn 2 + β ln Fn 4 We obtain the following solution of the ODE system 23) F l 3 F l + d l F 5 l = 0 23) F l = κ l G l = 2a lµ 1 µ 3 b l µ 2 µ 3 µ 2 )ξ + C l 24) 2µ 1 a l µ 1 b l µ 2 ) where C l, κ l, µ 1, µ 2, µ 3,a l, and b l are arbitrary constants that satisfy the following algebraic equations 4a l µ 2 1κ 4 l d l + 4a l µ 2 1κ 2 l c l 4a l µ 1 µ 2 µ 3 4µ 1 µ 2 κ 2 l b lc l 4µ 1 µ 2 κ 4 l b ld l + b 2 l µ2 2µ b l µ 2 2µ 3 + µ a l µ 2 1 α ln κ 2 n 4µ 1µ 2 b l α ln κ 2 n + 4a lµ 2 1 γ ln κ 2 n 4µ 1 µ 2 b l κ2 l γ ln κ 2 n κ2 l + 4a lµ 2 1 β ln κ 4 n 4µ 1µ 2 b l β ln κ 4 n = 0 25) The corresponding solution of the system of equations 2) is given by { [ q l) x,t) = κ l exp i µ 3 x 2a lµ 1 µ 3 b l µ 2 µ 3 µ 2 )µ 1 x µ 2 t) + C l ]}, 26) 2µ 1 a l µ 1 b l µ 2 ) where 1 l N. 4. CONCLUSIONS In this paper, we have considered the dynamics of N-coupled NLSEs with spatio-temporal dispersion by using the Lie symmetry analysis. Two types of nonlinear media that have been considered are those with Kerr law nonlinearity and parabolic law nonlinearity. The systems of equations 1) and 2) under consideration was analyzed by the theory of Lie symmetry method to reduce it to ordinary differential equations. We note that because of the dimensions of the Lie algebras, we did not resort to optimal symmetry technique. Corresponding to each reduction, certain exact solutions of the nonlinear partial differential equations are obtained. c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948

11 11 Lie symmetry analysis and exact solutions to N-coupled NLSEs Article no. 103 These solutions are very useful in the soliton community especially in the field of nonlinear optics and plasma physics where different modifications of the standard NLSEs frequently arise. The novelty of this work lies in the fact that the application of Lie symmetry analysis is being made for the first time in such systems, to the best of our knowledge. This integration algorithm has been so far applied only to polarization-preserving fibers and birefringent fibers [20]. In future, this study will be extended further along. The study of N-coupled NLSEs with four-wave mixing terms will be conducted later. The Lie symmetry analysis will expose solitons and other exact solutions that will give a broader perspective in this area of research. The obtained results will be reported elsewhere. Acknowledgements. The work of the fourth author QZ) was supported by the National Science Foundation for Young Scientists of Wuhan Donghu University. The fifth author SPM) would like to thank the research support provided by the Department of Mathematics and Statistics at Tshwane University of Technology and the support from the South African National Foundation under Grant Number IRF The research work of eighth author MB) was supported by Qatar National Research Fund QNRF) under the grant number NPRP REFERENCES 1. M. Mirzazadeh, M. Eslami, M. Savescu, A. H. Bhrawy, A. A. Alshaery, E. M. Hilal, E. M., and A. Biswas, Optical solitons in DWDM system with spatio-temporal dispersion, Journal of Nonlinear Optical Physics & Materials, 24, ). 2. M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, Society for Industrial and Applied Mathematics, Philadelphia, USA, R. Hirota, The direct method in soliton theory, Cambridge University Press, Cambridge, UK, C. Rogers and W. K. Schief, Bäcklund and Darboux transformations: geometry and modern applications in soliton theory, Cambridge University Press, Cambridge, UK, W. X. Ma, T. Huang, and Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application, Physica Scripta 82, ). 6. A. Biswas, M. Mirzazadeh, M. Savescu, D. Milovic, K. R. Khan, M. F. Mahmood, and M. Belic, Singular solitons in optical metamaterials by ansatz method and simplest equation approach, Journal of Modern Optics, 61, ). 7. A. J. M. Jawad, M. D. Petković, and A. Biswas, Modified simple equation method for nonlinear evolution equations, Applied Mathematics and Computation 217, ). 8. P. J. Olver, Applications of Lie groups to differential equations, Springer Science & Business Media, G. Bluman and K. Sukeyuki, Symmetries and differential equations, Springer Science & Business Media, Germany, N. H. Ibragimov, Transformation groups applied to mathematical physics, Springer Science & Business Media, Germany, S. Kumar, Q. Zhou, A. H. Bhrawy, E. Zerrad, A. Biswas, and M. Belic, Optical solitons in birefringent fibers by Lie symmetry analysis, Romanian Reports in Physics 68, ). 12. D. Mihalache, Multidimensional localized structures in optical and matter-wave media: A topical survey of recent literature, Romanian Reports in Physics, 69, ). c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948

12 Article no. 103 Yakup Yıldırım et al S. Chen, P. Grelu, D. Mihalache, and F. Baronio, Families of rational soliton solutions of Kadomtsev-Petviashvili I equation, Romanian Reports in Physics 68, ). 14. Y. B. Liu, A. S. Fokas, D. Mihalache, and J. S. He, Parallel line rogue waves of the third-type Davey-Stewartson equation, Romanian Reports in Physics 68, ). 15. J. S. He et al., Handling shocks and rogue waves in optical fibers, Romanian Journal of Physics 62, ). 16. A. Biswas and C. M. Khalique, Optical quasi-solitons by Lie symmetry analysis, Journal of King Saud University - Science 24, ). 17. M. Savescu, A. H. Bhrawy, E. M. Hilal, A. A. Alshaery, and A. Biswas, Optical solitons in birefringent fibers with four-wave mixing for Kerr law nonlinearity, Romanian Journal of Physics, 59, ). 18. Q. Zhou, Q. Zhu, L. Moraru, and A. Biswas, Optical solitons with spatiallly dependent coefficients by Lie symmetry, Optoelectronics and Advanced Materials - Rapid Communications 8, ). 19. S. Kumar, M. Savescu, Q. Zhou, A. Biswas, and M. Belic, Optical solitons with quadratic nonlinearity by Lie symmetry analysis, Optoelectronics and Advanced Materials - Rapid Communications 9, ). 20. S. Kumar, Q. Zhou, A. Biswas, and M. Belic, Optical solitons in nano-fibers with Kundu-Eckhaus equation by Lie symmetry analysis, Optoelectronics and Advanced Materials - Rapid Communications 10, ). 21. Y. S. Zhang, L. J. Guo, A. Chabchoub, and J. S. He, Higher-order rogue wave dynamics for a derivative nonlinear Schrödinger equation, Romanian Journal of Physics 62, ). 22. Y. S. Zhang, D. Q. Qiu, Y. Cheng, and J. S. He, Rational solution of the nonlocal nonlinear Schrödinger equation and its application in optics, Romanian Journal of Physics 62, ). 23. Q. Zhou, Q. Zhu, M. Savescu, A. Bhrawy, and A. Biswas, Optical solitons with nonlinear dispersion in parabolic law medium, Proceedings of the Romanian Academy, Series A 16, ). 24. E. V. Krishnan, Q. Zhou, and A. Biswas, Solitons and shock waves to Zakharov-Kuznetsov equation with dual-power-law nonlinearity in plasmas, Proceedings of the Romanian Academy, Series A 17, ). c) RJP 63Nos. 1-2), id: ) v.2.1* #b79a7948