SOLITON SOLUTIONS OF THE CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION WITH VARIABLE COEFFICIENTS

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1 SOLITON SOLUTIONS OF THE CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION WITH VARIABLE COEFFICIENTS HOURIA TRIKI 1, ABDUL-MAJID WAZWAZ 2, 1 Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, Annaba, Algeria trikihouria@gmail.com 2 Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA Corresponding author wazwaz@sxu.edu Received January 4, 2016 In this work, a nonlinear Schrödinger equation with variable coefficients incorporating cubic-quintic nonlinearities, self-steepening, and self-frequency shift is investigated by application of the trial equation method. The model describes the propagation of femtosecond light pulses in optical fibers. Exact soliton-like solutions including bright, dark, and singular solutions are derived. Parametric conditions for the existence of envelope soliton solutions are given. Key words: Trial equation method; cubic-quintic Schrödinger equation; Travelling wave solutions. 1. INTRODUCTION Nonlinear propagation of solitons has drawn considerable attention in a range of physical settings, including fluid dynamics and plasma physics [1], atomic physics and Bose-Einstein condensates [2]-[7], nonlinear optics and photonics [8]-[16], and so on. Solitons are defined as localized waves that propagate without change of their shape and velocity properties and are stable against mutual collisions [17]. The distinction between solitary wave and soliton solutions is that when any number of solitons interact they do not change form, and the only outcome of the interaction is a phase shift [18]. The existence of solitary wave solutions implies perfect balance between nonlinearity and dispersion, which usually requires rather specific conditions and cannot be established in general [19]. As well known, the nonlinear Schrödinger NLS) equation is a generic model for describing the dynamics of light pulses in optical fibers [8]. For picosecond light pulses, the NLS equation includes only the group velocity dispersion GVD) and the self-phase modulation, well known in fibers, and it admits bright and dark solitontype pulse propagation in anomalous and normal dispersion regimes, respectively [20]. However, as one increases the intensity of the incident light power to produce shorter femtosecond) pulses, additional nonlinear effects become important and the dynamics of pulses needs to be described in the frame of a generalized NLS equation that includes higher-order nonlinear terms [21]. RJP Rom. 61Nos. Journ. Phys., 3-4), Vol , Nos. 3-4, 2016) P , c) 2016 Bucharest, - v.1.3a*

2 2 Soliton solutions of the cubic-quintic nonlinear Schrödinger equation 361 The investigation of exact soliton-like solutions to nonlinear wave equations is of great value in understanding widely different physical phenomena. Recently, many powerful methods such as the sine-cosine methods [22]-[24], the subsidiary ordinary differential equation method [25]-[27], the Hirota s method [28], the Petrov- Galerkin method [29], the collocation method [30], the solitary wave ansatz method [31, 32], the Exp-function method [33], the trial equation method [34, 35], and many others, have been successfully applied to exactly solve nonlinear wave equations with constant and variable coefficients. Recently, Liu [36] proposed a trial equation method which can be suitable to both real equations and complex equations with variable coefficients. Liu [36] used this method to obtain some exact envelope traveling wave solutions for the timevarying NLS equation with GVD and Kerr nonlinearity. Subsequently, the same method has been applied to construct envelope traveling wave solutions of the generalized NLS equation with time-dependent coefficients involving an external potential term in addition to GVD and nonlinearity terms [37]. The trial equation method is a useful systematic method capable of solving nonlinear equations by using linear methods. In this paper, the trial equation method will be extended to cubic-quintic NLS equation with self-steepening and self-frequency shift [38]: iq t + ft)q xx + gt) q 2 + σ q 4) q = iht) q 2 q )x + ipt) q 2) q, 1) x where ft), gt), ht), and pt) are time-dependent functions. In Eq. 1), qx,t) is the complex envelope of the electric field, ft) is the timedependent dispersion coefficient, ht) is the self-steepening coefficient, and pt) is the self-frequency shift coefficient. The third and fourth terms represent cubic and quintic nonlinearities, respectively, σ is a constant, and the subscripts x and t denote the spatial and temporal partial derivatives, respectively. Recently, Green and Biswas [38] studied Eq. 1) by the ansatz method and obtained the exact one-soliton solution under certain parametric conditions. To the best of our knowledge, no attempt was made regarding soliton-like solutions of the NLS equation with parabolic law nonlinearity and variable coefficients 1) by using the trial equation method. Here, we investigate the applicability and effectiveness of the trial equation method that was recently proposed by Liu [36] on the cubicquintic NLS equation with time-dependent coefficients. We will show that a subtle interplay between the group velocity dispersion, self-steepening, self-frequency shift, and cubic-quintic nonlinearities, can result in a rich variety of shape-preserving waves with interesting properties.

3 362 Houria Triki, Abdul-Majid Wazwaz 3 2. EXACT SOLUTIONS TO EQUATION 1) To seek the envelope traveling wave solutions of 1), we assume the solution in the form [36] qx,t) = uξ)e iζ, ξ = kt)x + ω t), ζ = st)x + rt). 2) Here u is a function of ξ to be determined, and kt), ω t), st), and rt) are timedependent parameters that will be also determined. Substituting Eq. 2) into Eq. 1) and separating the real and imaginary parts leads to [ k t)x + ω t) + 2ft)st)kt) ] u [3ht) + 2pt)]k t)u 2 u = 0, 3) ft)k 2 t)u [ s t)x + r t) + ft)s 2 t) ] u + [gt) + ht)st)]u 3 +σgt)u 5 = 0. 4) Now we suppose that the exact solution of Eq. 1) satisfies the following trial equation [36] u ) 2 m = F ξ) = a i u i, 5) where a i i = 0,...,m) are constants and m is an integer to be determined later. Considering homogeneous balance between u 5 and u in Eq. 4), we can determine the value of m in Eq. 5) as m = 6. Inserting the resulting trial Eq. 5) into 4), and setting each coefficients of u, u 2 u, and u i ξ) with i = 0,...,6) in Eqs. 3) and 4) to zero yields i=0 k t)x + ω t) + 2ft)st)kt) = 0, 6) [3ht) + 2pt)]k t) = 0, 7) a 2 s t)x + r t) + ft)s 2 t) ft)k 2 = 0, t) 8) gt) + ht)st) 2a 4 + ft)k 2 = 0, t) 9) 3a 6 σ + gt) ft)k 2 = 0, t) 10) a 1 = 0, a 3 = 0, a 5 = 0, 11)

4 4 Soliton solutions of the cubic-quintic nonlinear Schrödinger equation 363 Solving Eqs. 6)-11) yields kt) = k, st) = s, ωt) = 2sk ft)dt, rt) = c 1 ft)dt, 12) gt) = c 2 k 2 ft), ht) = c 3 k 2 ft), pt) = 3 2 c 3k 2 ft), 13) a 4 = c 2 + sc 3, a 2 = c 1 + s 2 2 k 2, a 6 = σc 2 3, 14) a 1 = a 3 = a 5 = 0, a 0 = c 4, 15) where k, s, c 1, c 2, c 3, and c 4 are arbitrary constants. Under the conditions 15), Eq. 5) with m = 6 is reduced to the following expression u ) 2 = a0 + a 2 u 2 + a 4 u 4 + a 6 u 6, 16) The integral form of Eq. 16) is 1 ±ξ ξ 0 ) = du. 17) a0 + a 2 u 2 + a 4 u 4 + a 6 u6 In what follows, we discuss the case where a 0 = 0. Then equation 17) is reduced to the following form 1 ±ξ ξ 0 ) = u du. 18) a 2 + a 4 u 2 + a 6 u4 Denote = a 2 4 4a 2a 6, according to Yomba [39], we obtain four families of soliton solutions: Family 1: = a 2 4 4a 2a 6 > 0 and a 2 > 0. Then the exact solution of Eq. 1) is a bright-soliton-type solution of the form [ ] 1/2 2a 2 u 1 = ɛ cosh 2 a 2 ξ ) expiζ). 19) a 4 Family 2: = a 2 4 4a 2a 6 < 0 and a 2 > 0. Then one obtains an exact singularsoliton-type solution for Eq. 1) as [ ] 1/2 2a 2 u 2 = ɛ sinh 2 a 2 ξ ) expiζ). 20) a 4 Family 3: = a 2 4 4a 2a 6 = 0 and a 2 > 0. Then Eq. 1) has a dark-solitontype solution given by { u 3 = a [ )]} 1/2 2 a2 1 + ɛtanh a 4 2 ξ expiζ). 21)

5 364 Houria Triki, Abdul-Majid Wazwaz 5 Family 4: a 2 > 0. We obtain another bright-type soliton-like solution for Eq. 1) as { a 2 a 4 sech 2 a2 ξ ) } 1/2 u 4 = a 2 4 a [ 2a ɛtanh a2 ξ )] 2 expiζ}, 22) where ɛ = ±1. Note that solutions 19)-22) are all possible solutions to Eq. 1) in the case a 6 0. It is significant to observe that the obtained solutions exist under the parametric condition 3ht) + 2pt) = 0. The later means that self-steepening and selffrequency shift coefficients are not independent and the existing soliton solutions are obtained in the framework of this relationship. 3. CONCLUSIONS In this paper, we have investigated a higher-order nonlinear Schrödinger equation with time-dependent coefficients, modeling the propagation of ultrashort femtosecond) optical pulses in nonlinear optical fibers. The model used combines cubic and quintic nonlinearities, as well as the self-steepening and self-frequency shift effects. The trial equation method is used to construct families of bright, dark, and singular soliton solutions for the nonlinear dynamical model. Conditions for the existence of propagating envelope solutions have also been reported. The work reveals the power of the trial equation method in handling nonlinear evolution equations with variable coefficients. The trial equation method is a useful systematic method capable of solving nonlinear evolution equations by using linear methods. REFERENCES 1. P. K. Shukla and A. A. Mamun, Solitons, shocks and vortices in dusty plasmas, New J. Phys. 5, ). 2. S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Dark solitons in Bose-Einstein condensates, Phys. Rev. Lett. 83, ). 3. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Formation of a matter-wave bright soliton, Science 296, ). 4. K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Formation and propagation of matter-wave soliton trains, Nature 417, ). 5. A. I. Nicolin, A. Balaz, J. B. Sudharsan, and R. Radha, Ground state of Bose-Einstein condensates with inhomogeneous scattering lengths, Rom. J. Phys. 59, ). 6. V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, Bose- Einstein condensation: Twenty years after, Rom. Rep. Phys. 67, ). 7. A. I. Nicolin, M. C. Raportaru, and A. Balaz, Effective low-dimensional polynomial equations for Bose-Einstein condensates, Rom. Rep. Phys. 67, ). 8. G. P. Agrawal, Nonlinear Fiber Optics 4th edn., Academic Press, New York, 2007).

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