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single-mode optical fibers. Carrying out numerical computation to a model of coupled NLSE, the analysis of pulse evolution shows that the Kerr effect stabilizes solitons against spreading caused by birefringence. Furthermore, the study concluded that the birefringence must be linear and the standard single-mode birefringent fibers are much suitable for soliton communication system. Recently, Liu et al. [ 20 ] investigated the optical nondegenerate solitons in a birefringent fibers with 35 ° elliptical angle. Making use of the bilinear forms, the nondegenerate bright oneand two-soliton solutions are constructed by solving the coupled NLSE. By virtue of the asymptotic analysis, two degenerate solitons are found to be elastic interactions under certain conditions while two nondegenerate solitons are elastic interactions but it can be made inelastic by adjusting the soliton phase. In an inhomogeneous birefringent nonlinear dispersive medium, Zaabat et al. [ 21 ] scrutinized wave-speed management of soliton pulses in emergence of variable dispersion and nonlinearity parameters and external potential. New types of periodic nonlinear waves expressed in the form of Jacobi elliptic functions are extracted which degenerate, as the elliptic modulus approaching unity, to soliton pulse solutions including bright-dipole and bright-dark solutions. It is noted that the wave speed of solitons can be controlled by varying dispersion parameter. In addition, the influence of fractional order derivatives on optical solitons transmitted through birefringence fibers are also reported by different groups of authors. Applying the semi-inverse and fractional variational method, Fu et al. [ 22 ] discussed the space-time fractional NLSE with different types of nonlinearity such as Kerr, power, parabolic, dual-power, and log law. Their study results in miscellaneous soliton structures including bright, dark, and singular solitons which display diverse behaviors because of fractional order effects. Considering conformable fractional derivative, Li et al. [ 23 ] dealt with Fokas-Lenells equation with the aid of the plane dynamics system. The optical soliton solutions, qualitative analysis and chaotic behaviors of the model are illustrated thoroughly.
Moreover, the effects of the phase chirp on transmitted solitons cannot be ignored as this chirp can rehash the physical aspects of propagating pulses. Mahmood [ 24 ] studied the behavior of chirped solitons in a single-mode birefringent fiber described by a coupled NLSE. It is detected that the chirp has a significant role on controlling the threshold amplitude for soliton trapping without causing excessive pulse broadening. Besides, Triki et al. [ 25 ] investigated ultrashort light pulses in a birefringent optical fiber under the effect of several physical features. The chirped solitons in the form of dark-dark and bright-bright soliton pairs are induced in the presence of all fiber parameters. The consequence of study indicates that the frequency chirp corresponding to these solitons depends on the intensity of the pulse. Using numerical simulation scheme, Xiao et al. [ 26, 27 ] discussed chirped bright and dark vector quasi-solitons in birefringent fiber system characterized by the coupled Ginzburg-Landau equation. The analyzed behaviors displays that both bright and dark vector solitons can be transmitted stably in the birefringent fiber system.
For more details about the analytical studies of soliton propagation in birefringent fiber, the reader is referred to the references [ 28 – 40 ].
As mentioned above, the dynamics of light pulses transmitted through nonlinear optical media are described by the NLSE-type equations and relevant models. Kaup- Newell equation( KNE) is one of the vital models that describes wave propagations in optical fiber and plasma physics [ 41 – 44 ]. This model is found to characterize the sub-pico-second pulses in mono-mode optical fibers. The dimensionless form of KNE is given by [ 45, 46 ]
! t þ ia! xx þ bðj! j 2! Þ x
¼ 0; ð1Þ
where!( x, t) represents a complex-valued function referring to the wave profile. The parameter a denotes the group velocity dispersion while the parameter b describes the effect of nonlinearity in the medium. Equation( 1) has been scrutinized in the past to derive the exact soliton solitons. Two group of authors [ 47, 48 ] investigated the chirped optical solitons using different integration methodologies. Numerous types of structures are obtained such as bright, dark and singular solitons. The chirped solitons are extracted with their corresponding chirp which is expressed nonlinearly in terms of soliton intensity. With the aid of a novel mathematical technique, new chirped optical soliton solutions are retrieved which can be useful in the fields of optical fibers and plasma physics [ 49 ]. Further to this, the perturbed model of KNE has been studied in the past to detect soliton pulses, see [ 50, 51 ].
The KNE can be applied in birefringent fiber without four-wave mixing to study the dynamics of optical solitons and the physical properties of the medium. Thus, the model of KNE is addressed as
p t þ ia 1 p xx þ b 1 ðjpj 2 pÞ x þ c 1 ðjqj 2 qÞ x
¼ 0; q t þ ia 2 q xx þ b 2 ðjqj 2 qÞ x þ c 2 ðjpj 2 pÞ x
¼ 0; ð2Þ
where a j( j = 1, 2) represents the coefficients of group velocity dispersion while b j and c j( j = 1, 2) stand for the nonlinear influence. The previous studies in literature that scrutinized the coupled equations( 2) have investigated the chirped-free solitons only. For example, three groups of authors [ 52 – 54 ] studies sub pico-second optical pulses of equations( 2) by applying distinct mathematical tools. Different wave structures are extracted including bright, dark and singular solitons. Exploiting three integration approaches, the system of KNE( 2) is revisited by Rehman et al. [ 55 ] to investigate the sub-pico-second optical solitons. The detected wave forms include bright, dark, singular and bright-singular combo solitons as well as periodic singular waves. Recently, Li [ 56 ] discussed the dynamical behavior of the coupled KNE by means of bifurcation theory of planar dynamical system. Using this technique, the phase portrait and optical soliton solutions are created.
Our present attention is concentrated on investigating the chirped bright and dark optical pulses of fractional KNE in birefringent fiber in the sense of conformable