Extraction of optical solitons in birefringent bers for Biswas-Arshed equation via extended trial equation method

Abstract: This article obtains optical solitons to the Biswas-Arshed equation for birefringent bers with higher order dispersions and in the absence of four-wave mixing terms, in amediawithKerr type nonlinearity. Optical dark, singular and bright soliton solutions are articulated by applying an imaginative integration technique, the extended trial equation scheme. Various additional traveling wave solutions are produced with this integration technique, which include rational solutions, Jacobi elliptic function solutions and periodic singular solutions. From the mathematical analysis some constraints are recognized that ensure the actuality of solitons.


Governing equation
The coupled system obtained from (1) in birefringent bers without 4WM reads [46] ip t + a pxx Here b and a are the coe cients of spatio-temporal dispersion (STD) and group velocity dispersion respectively, while d and c are the coe cients of third order STD and third order dispersion respectively, for = , . Next, γ and λ stand for self-steepening terms, while the nonlinear dispersions are con rmed by β , θ , α and µ .
Here Q (ζ ) is the amplitude, ν gives the soliton velocity, ϵ is the phase constant, κ and ω are respectively the frequency and wave number of the soliton. Implanting Eqs. (3)-(5) into (2) and dividing into imaginary and real parts, we attain and where * = − and = , . The balancing principle suggests that Q l * = Q . Therefore, from Eq. (6) we have along with the restraint conditions Comparing the two values of ν in (9), leads to the restraints Now, from Eq. (7), we have where = , . Eq. (12) will now be scrutinized by extended trial equation method.

. Application of extended trial equation method
In this subsection, we employ the extended trial equation technique [7,8] to Eq. (12) for constructing the exact solutions of the system (2). Case-1. The solution of Eq. (12) can be expressed as where δ i are unknown constant to be determined such that δϱ ≠ and where η , ..., ησ and χ , ..., χρ are arbitrary constants to be identi ed such that ησ ≠ and χρ ≠ . Eq. (14) can be transformed into an integral form as: The balancing process reveals that σ = ρ + ϱ + .
By assuming σ = , ϱ = and ρ = in (16), we arrive at Implanting Eq. (17) along with Eq. (14) into (12) and evaluating the resultant system of equations, we attain where Substituting the values of parameters from (18) into (14) and using Eq. (15), we obtain where As a consequence, the following exact solutions can now be written for the coupled system (2). If and Whenever where and υ j , j = , . . . , are the roots of Λ(Ψ) = .
By assuming δ = −δ υ and ζ = , the solutions given by (23)- (32) can be transformed to the plane wave solutions singular soliton solutions bright soliton solutions Moreover, when δ = −δ υ and ζ = , Jacobi elliptic function solutions (33) and (34) trimmed as Remark 1. When the modulus m → , singular soliton solution emerge as where υ = υ . Remark 2. When the modulus m → , periodic singular solution emerge as where υ = υ . Case-2. Eq. (12) can be written as through the transformation Q = P / . Therefore, the solution of Eq. (50) can be expressed as where δ i are unknown constant to be determined such that δϱ ≠ . The balancing process reveals that σ = ρ + ϱ + .
By assuming σ = , ρ = and ϱ = in Eq. (52), we arrive at Implanting Eq. (53) along with Eq. (14) into Eq. (50) and evaluating the resultant system of equations, we attain where Substituting all the values from (54) into (14) and using Eq. (15), we obtain where As a consequence, the following exact solutions can now be written for the coupled system (2).

Conclusion
The work expounded in this article successfully that addresses optical solitons of Biswas-Arshed equation with Kerr-law nonlinearity in birefringent bers with higher order dispersions and in the absence of fourwave mixing terms by the application of extended trial equation technique. With this integration scheme, we have commendably recovered dark, bright and singular optical solitons along with other traveling wave solutions, comprising rational solutions, periodic singular solutions and Jacobi elliptic function solutions in the presence of some constraints. It is concluded that our derived results for the Biswas-Arshed equation in birefringent bers are exclusively new and have not been stated earlier.
The outcomes of this paper are attention-grabbing and provide a stimulus to the audience of optical solitons. Later, this equation will be studied with the addition of four-wave mixing terms by the aid of appreciated integration schemes. These precious results will be presented as soon as possible.

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Con ict of interest:
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