Bright and Dark Solitons in Optical Fibers with Parabolic Law Nonlinearity

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1 SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vo. 0, No. 3, October 03, UDK: DOI: 0.98/SJEE M Bright Dark Soitons in Optica Fibers with Paraboic Law Noninearity Daniea Miović, Anjan Biswas Abstract: This paper utiizes the ansatz method to obtain bright dark -soiton soution to the noninear Schrodinger s equation with paraboic aw noninearity in birefringent fibers. There are a few Hamitonian type perturbation terms taken into account. The eact soiton soution comes with baggages that are referred to as constraint conditions that must hod in order for these soitons to eist. Keywords: Birefringence; Integrabiity; Soitons. Introduction Optica soitons is one of the most important topics of research in noninear fiber optics [ 0]. Soitons form the basic fabric of fiber optic communications across trans-continenta trans-oceanic distances. It is therefore imperative to take a deeper ook at optica soitons from a different perspective. This paper wi study the dynamics of optica soitons when the optica fiber maintains cubic-quintic aw of noninearity that is aso known as the paraboic aw noninearity. The focus wi be on etracting eact bright dark -soiton soution in birefringent fibers in presence of perturbation terms. Whie the norm is to ook into optica fibers with Kerr aw noninearity, this paper studies paraboic aw fibers so that the resuts wi st on a generaized setting as compared to Kerr aw that was studied earier []. The ansatz approach wi be the integration too appied to obtain eact -soiton soution to the governing couped noninear Schrodinger s equation (NLSE) with paraboic aw noninearity in birefringent fibers. Using this approach both bright dark soiton soutions wi be obtained. There are severa constraint conditions that wi fa out. These constraints wi be necessary conditions for bright dark soitons to eist. The perturbation terms that wi be taken into account are inter-moda dispersion, noninear dispersion sef-steepening. These are a Hamitonian Facuty of Eectronic Engineering, Department of Teecommunications, University of Niš, Aeksra Medvedeva 4, 8000 Niš, Serbia; E-mai: daniea.miovic@efak.ni.ac.rs Department of Mathematica Sciences, Deaware State University, Dover, DE , USA; E-mai: biswas.anjan@gmai.com 365

2 D. Miović, A. Biswas type perturbations [4, 6, 7, 9]. Therefore the integrabiity aspect of the governing NLSE wi not be hampered. Governing Equation The governing equation for the propagation of soitons through optica fibers is the NLSE. However, in presence of birefringence when the puse spits into two parts, the corresponding mode is the couped NLSE. For paraboic aw noninearity, in birefringent fibbers, this couped equation reads t iq a q c q d r q q q r r q t i q q q q q q q 0, ir a r c r d q r r r q q r i r r r r r r r 0. In () () q(, t) r(, t) are compe vaued functions that represent the soiton profies for the two components in birefringent fibers. For =,, a represents the group veocity dispersions GVD, c d represents the sef-phase moduation (SPM) cross-phase moduation (XPM) terms respectivey. From the perturbation terms α represents the inter-moda dispersion, λ is the sef-steepening terms to avoid the formation of shock waves, ν θ are noninear dispersions. The terms with, are associated with the quintic terms of the cubic-quintic aw of noninearity. In order to obtain an eact bright dark -soiton soution to previous equations we use the ansatz method. At the starting point, the soitons are considered in the phase-ampitude format as [6, 7] ( t, ) i( t) qt (, ) Pte (, ) Pte (, ), (3) ( t, ) i( t) rt (, ) P( te, ) P( te, ), (4) where P for =, are the ampitude components of the soitons (, t) are its phase components, that are defined as ( t, ) t. (5) Here κ are the frequencies of the soitons in two components, ω are the wave numbers, whie σ are the phase constants. Substituting (3) (4) into () () decomposing into rea imaginary parts ead to () () 366

3 Brigth Dark soitons in optica fibers with paraboic aw noninearity ( a ) P d PP P P P PP P ( c) P a 0 P P P t respectivey for =, 3. (a 3 ) (3 ) P 0 The rest of the section wi study two different types of soitons, namey bright soitons dark soitons. The ansatz approach wi be our integration architecture.. Bright soitons For bright soitons, the assumption is [] A P (, t) p ( D cosh ), (8) where B( vt). (9) Here, A ( =, ) represents the ampitude of the soitons for the two components B is the inverse width of the soitons in both components v is the speed of the soitons in both components. The two new parameters introduced are D for =,. Substituting this hypothesis into the imaginary part equation (7) eads to (3 ) A ( v a ) 0. p D cosh (0) Setting the coefficients of the ineary independent functions from (0) impies v a () 3 0. () Equation () is the veocity of the soiton for two components equation () represents the constraint condition in order for the soiton to eist. From () equating the veocity of the soitons in the two components eads to the constraint condition given by a a. (3) Net, the rea part equation reduces to (6) (7) 367

4 D. Miović, A. Biswas ap(p ) DB ap(p )( D ) B cosh ( a p ab ) D cosh D 4 ca da A D cosh D cosh D cosh p p 4p 4 AA A 0. p p D cosh D cosh D cosh p (4) By using baancing principe, equating the eponents ( p p ) gives p p. (5) Setting the coefficients of the ineary independent functions to zero eads to ab 4 ( a ), (6) 4 B c A Da 4 A, (7) 3Ba 4 D B 3 a, (8) which poses the constraint condition Da ( c) 0 (9) a (3B a 4 4 A ) 0 (0) for =,. Hence, finay the bright -soiton soution for paraboic aw, in birefringent fibers is given by qt (, ) A e D cosh B( vt) i( t) () A i( t) rt (, ) e, () D cosh B( vt) where the parameter specifications are just discussed in detais above.

5 Brigth Dark soitons in optica fibers with paraboic aw noninearity. Dark soitons For dark soitons, the assumption is [8] P(, t) ( A B tanh ), (3) where the is defined as in (9). Here, A B are free parameters. Substituting this hypothesis into the imaginary part equation (7) eads to B ( tanh ) v a A B (4) p (3 )( A B tanh ) 0. From the ineary independent functions, one recovers (3) 3 0, (5) which is a constraint condition between the parameters. Then, the rea part equation given by (6) simpifies to 3 B pa ( p )( p )( A B tanh ) B Apa ( p )( A B tanh ) B a p B a(3 A B ) ( A B tanh ) p(p ) B Aa ( B A )( A B tanh ) p B p( p ) a( B A ) ( c)( A B tanh ) (6) p 4p d ( A Btanh ) ( A Btanh ) ( A Btanh ) p p ( A B tanh ) ( A B tanh ) 4 p A B A B ( tanh ) ( tanh ) 0. The baancing principe yieds the same vaue of p as given by (5) for bright soitons. The other parameter vaues that are obtained from the rea part are 3 ab 8 8a 9a, (7) 8 c d B a, (8) 0, (9) A B. (30) Hence, finay, the dark -soiton soution in birefriengent fibers is given by i( t) i( t) qt (, ) A tanh B ( vt) e, (3) rt (, ) A tanh B ( vt) e. (3) 369 p 3

6 D. Miović, A. Biswas These are the eact dark -soiton soutions of the two components of the soitons in a birefringent fiber with paraboic aw noninearity. The parameter definitions the necessary constraints are in pace. 3 Concusion This paper obtained the bright dark -soiton soution to the NLSE with paraboic aw noninearity. Severa constraint conditions are in pace in order for the soitons to eist. These constraints are necessary conditions. In future, one can etend these resuts to the case of Thirring soitons, DWDM systems other aspects. The resuts of those researches wi be reported in future. These form a tip of the iceberg. 4 References [] A. Biswas, S. Konar: Introduction to non-kerr Law Optica Soitons, CRC Press, Boca Raton, FL, USA, 006. [] A. Biswas, K. Khan, A. Rahman, A. Yidirim, T. Hayat, O.M. Adossary: Bright Dark Optica Soitons in Birefringent Fibers with Hamitonian Perturbations Kerr Law Noninearity, Journa of Optoeectronics Advanced Materias, Vo. 4, No. 7-8, Juy/Aug. 0, pp [3] Z. Jovanoski, D.R. Row: Variationa Anaysis of Soitary Waves in a Homogeneous Cubic-quintic Noninear Medium, Journa of Modern Optics, Vo. 48, No. 7, June 00, pp [4] R. Koh, A. Biswas, D. Miovic, E. Zerrad: Optica Soiton Perturbtion in a non-kerr Law Media, Optics Laser Technoogy, Vo. 40, No. 4, June 008, pp [5] L. Lenes, A.S. Fokas: Dressing for a Nove Integrabe Generaization of the Noninear Schrodinger's Equation, Journa of Noninear Science, Vo. 0, No. 6, Dec. 00, pp [6] M. Savescu, K.R. Khan, P. Naruka, H. Jafari, L. Moraru, A. Biswas: Optica Soitons in Photonic Nanowaveguides with an Improved Noninear Schrodinger's Equation, Journa of Computationa Theoretica Nanoscience, Vo. 0, No. 5, May 03, pp [7] M. Savescu, K.R. Khan, R.W. Koh, L. Moraru, A. Yidirim, A. Biswas: Optica Soiton Perturbation with Improved Noninear Schrodinger's Equation in Nano Fibers, Journa of Nanoeectronics Optoeectronics, Vo. 8, No., Feb. 03, pp [8] H. Triki, A. Biswas: Dark Soitons for a Generaized Noninear Schrodinger's Equation with Paraboic Dua-power Law Noninearities, Mathematica Methods in the Appied Sciences, Vo. 34, No. 8, May 0, pp [9] E. Topkara, D. Miovic, A.K. Sarma, E. Zerrad, A. Biswas: Optica Soitons with non-kerr Law Noninearity Inter-moda Dispersion with Time-dependent Coefficients, Communications in Noninear Science Numerica Simuation, Vo. 5, No. 9, Sept. 00, pp [0] M. Wadati, H. Segur, M.J. Abowitz: A New Hamitonian Ampitude Equation Governing Moduated Wave Instabiities, Journa of the Physica Society of Japan, Vo. 6, No. 4, Apri 99, pp