Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 572 https://internationalpubls.com Application of the 𝑮’/𝑮 Expansion Method for Solving New Form of Nonlinear Schrödinger Equation in Bi-Isotropic Fiber Ourahmoun Abbes1, Mezache Zinelabiddine2 1, 2 Institute of Optics and Fine Mechanics, University of Ferhat Abbas Setif 1, Setif, Algeria. abbes.ourahmoun@univ-setif.dz zinemezaache@yahoo.fr Article History: Received: 21-04-2024 Revised: 19-06-2024 Accepted: 27-06-2024 Abstract: Bi-isotropic media (chiral and non-reciprocal) present an outstanding challenge for the scientific community. Their characteristics have facilitated the emergence of new and remarkable applications. In this paper, we focus on the novel effect of chirality, characterized through a newly proposed formalism, to highlight the nonlinear effect induced by the magnetization vector under the influence of a strong electric field. This research work is concerned with a new formulation of constitutive relations. We delve into the analysis and discussion of the family of solutions of the nonlinear Schrödinger equation, describing the pulse propagation in nonlinear bi-isotropic media, with a novel approach to constitutive equations. We apply the extended 𝐺’/𝐺-expansion method with varying dispersion and nonlinearity to define certain families of solutions of the nonlinear Schrödinger equation in bi-isotropic (chiral and non-reciprocal) optical fibers. This clarification aids in understanding the propagation of light with two modes of propagation: a right circular polarized wave (RCP) and a left circular polarized wave (LCP), each having two different wave vectors in nonlinear bi-isotropic media. Various novel exact solutions of bi-isotropic optical solitons are reported in this study. Introduction: The investigation of exact solutions for nonlinear partial differential equations (PDEs) holds significant importance in understanding nonlinear physical phenomena. Nonlinear waves manifest across various scientific domains, notably in optical fibers and solid-state physics. In recent years, several potent methodologies have emerged for identifying solitons and periodic wave solutions of nonlinear PDEs. These include the 𝐺’/𝐺 -expansion method [1-6], the new mapping method [9-10], the method of generalized projective Riccati equations [11-16], and the 𝐺’/𝐺 expansion method [17]. Consequently, an original mathematical approach is proposed to evaluate nonlinear effects in bi-isotropic optical fibers, stemming from magnetization under the influence of a strong electric field [19-20]. The extended 𝐺’/𝐺-expansion method emerges as a potent technique for deriving solution families of the nonlinear Schrödinger equation in bi-isotropic optical fibers. This method employs a perturbation expansion in powers of the dimensionless parameter and is applicable for both weak and strong nonlinearities. It accommodates varying dispersion and nonlinearity, rendering it suitable for modeling a wide array of optical fibers. Results and Conclusion: This investigate is concerned with a new formulation of constitutive relation linking to the magnetic effect, to understand rigorously the physical nature of biisotropic effects and to generalize the main macroscopic models. We inferred the nonlinear Schrodinger equation for a bi-isotropic medium term with a nonlinear term of magnetizing. In this article, the extended 𝐺’/𝐺-expansion method is a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 573 https://internationalpubls.com powerful technique for determining a family of solutions of the nonlinear Schrödinger equation in bi-isotropic optical fibers. This method is based on the use of a perturbation expansion in powers of the dimensionless parameter, and it is valid for both weak and strong nonlinearities. The method allows for the inclusion of varying dispersion and nonlinearity, making it well- suited for modeling a wide range of optical fibres. Overall, the extended 𝐺’/𝐺 -expansion method is a valuable tool for understanding the dynamics of nonlinear optical systems, and it is expected to have a wide range of applications in the field of nonlinear optics. Keywords: Nonlinear Bi-isotropic media , Schrödinger equation , Optical fiber ,The 𝐺’/𝐺 expansion method, chiratity, non-reciprocity. 1. Introduction The investigation of exact solutions for nonlinear partial differential equations (PDEs) holds significant importance in understanding nonlinear physical phenomena. Nonlinear waves manifest across various scientific domains, notably in optical fibers and solid-state physics. In recent years, several potent methodologies have emerged for identifying solitons and periodic wave solutions of nonlinear PDEs. These include the 𝐺’/𝐺 -expansion method [1-6], the new mapping method [9-10], the method of generalized projective Riccati equations [11-16], and the 𝐺’/𝐺 -expansion method [17]. Conte and Musette [11] introduced an indirect approach to uncover solitary wave solutions of specific nonlinear PDEs expressible as polynomials in two elementary functions satisfying a projective Riccati equation [18]. This method has been successfully applied to numerous nonlinear PDEs, with resulting solitary wave solutions documented in [12-16]. The 𝐺’/𝐺 -expansion method exhibits broad applicability for tackling various other nonlinear evolution equations in mathematical physics. In recent years, research endeavors have actively explored novel types of heterogeneous absorbent materials, notably bi-isotropic materials. Bi-isotropic materials comprise a random dispersion of inclusions within a polymeric or ceramic matrix. Lindman in 1920 and Pickering in 1945 specifically investigated the interaction of electromagnetic waves with a collection of randomly distributed metal helices of the same enantiomorph form [19-23]. They observed polarization plane rotation of electromagnetic waves post-interaction with the helices. In 1979, Jaggard, Mickelson, and Papas proposed a macroscopic model detailing the interaction of electromagnetic waves with chiral structures (bi-isotropic materials) [19-20]. The general bi-isotropic medium is chiral and non-reciprocal, involving a complex parameter in the general case. The effects of two parameters, nonlinear chirality, and non-reciprocity, are crucial for estimating non-linearity in bi-isotropic fibers. This paper examines a direct method to determine numerous solution families for the nonlinear Schrödinger equation in bi-isotropic optical fiber, alongside 𝐺’/𝐺-expansion. This contributes to a deeper understanding of the interaction between electromagnetic waves and nonlinear bi-isotropic media, facilitating the design of potential applications in microwave and optical domains. Consequently, an original mathematical approach is proposed to evaluate nonlinear effects in bi-isotropic optical fibers, stemming from magnetization under the influence of a strong electric field Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 574 https://internationalpubls.com [19-20]. The extended 𝐺’/𝐺 -expansion method emerges as a potent technique for deriving solution families of the nonlinear Schrödinger equation in bi-isotropic optical fibers. This method employs a perturbation expansion in powers of the dimensionless parameter and is applicable for both weak and strong nonlinearities. It accommodates varying dispersion and nonlinearity, rendering it suitable for modeling a wide array of optical fibers. The method effectively describes the propagation of optical pulses in bi-isotropic fibers and predicts their behavior under diverse conditions. Furthermore, it facilitates the study of soliton dynamics, including stability and interactions. Overall, the extended 𝐺’/𝐺 -expansion method stands as a valuable tool for comprehending the dynamics of nonlinear optical systems, with anticipated applications across the field of nonlinear optics [24]. 2. Description of the (𝐆′/𝐆)-expansion method Study a nonlinear PDE in the following form P(u, uz, ut, uzz, utt) = 0 (1) where u = u(t, z) is an new function, P is a polynomial in u = u(t, z) , and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. Let us now provide the main steps of the extended (G′/G)- expansion method . Step 1. We usage the following conversion u(z, t) = u(ξ); ξ = z − vt (2) to reduce Eq.(3) to the next nonlinear ODE: K(u, u′, u′′, . . . ) = 0 (3) where v is velocity of the propagation, K is a polynomial of u(ξ) and its derivatives u′(ξ), u ′′ (ξ), . . . where u′(ξ) = du ∂ξ , u′′(ξ) = d2u(ξ) dξ2 ,... Step 2. We adopt that the solution of Eq.(1) has the form: u(ξ) = a0(z) + ∑ N i=1 ai ( G′ G ) i + bi(z) ( G′ G ) i−1 √1 + 1 μ ( G′ G ) 2 (4) where ai = ai(z), bi = bi(z) (i = 1,2, . . N) are functions of z to be determined. G = G(ξ) satisfies the resulting second order ODE equation: G′′(ξ) + μG(ξ) = 0 (5) Where G′ = dG dξ , G′′ = d2G dξ2 . We describe the degree of u(ξ) as D[u(ξ)] = M , then D[ dpu(ξ) dξp ] = M + p ; D[up ( dqu(ξ) dξq ) s ] = Mp + (M + q)s (6) The parameter N can be start by balancing the highest order derivative term and nonlinear terms in Eq.(3). N is typically a positive integer. Substituting Eq.(1) along with Eq.(2) into Eq.(4), collecting all terms with the same power of ( G′(ξ) G(ξ) )i(i = 0; 1; 2;...N) , and setting each coefficients to zero, we get a system of algebraic equations for ai and bi.The solution of Eq.(5) are given as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 575 https://internationalpubls.com 3. G′(ξ) G(ξ) = { √−μ( A1sinh√−μξ+A2cosh√−μξ A1cosh√−μξ+A2sinh√−μξ ) ; μ < 0 √μ ( A1sin√μξ+A2cos√μξ A2sin√μξ−A1cos√μξ ) ; μ > 0 A2 A1+A2ξ ; μ = 0 (7) which can be written in the following simplified form G′(ξ) G(ξ) = { √−μtanh(√−μξ + ξ0); μ < 0, tanhξ0 = A1 A2 ; | A2 A1 | > 1 √−μcoth(√−μξ + ξ0); μ < 0, cothξ0 = A1 A2 ; | A2 A1 | < 1 √μcot(√μξ + ξ0); μ > 0, cosξ0 = − A2 A1 A2 A1+A2ξ ; μ = 0 (8) 4. By solving the over-determined algebraic system with the help of Mathematica, we obtain the values of a0; ai and bi. Substituting these results into Eq.(1), and combining with the solution of Eq.(5), we obtain some exact solutions of Eq.(1). 3. Application of the (𝐆′/𝐆)-expansion method in bi-isotropic fiber The nonlinear Schrödinger equation (ESNL) is an equation that occurs in several fields of wave physics [1-24]. Among other things, it can be obtained from an asymptotic expansion of the Korteweg- de Vries (KdV) equation for weakly nonlinear wave packets [1-5]. The Schrödinger equation then follows from a first order approximation and gives the evolution of the wave packet envelope [1-5]. The characterization of the bi-isotropic nonlinear medium is achieved through an electromagnetic approach utilizing a novel formulation of constitutive relations [22]. This formulation enables the derivation of the nonlinear Schrödinger equation for a chiroptic fiber, obtained through a first-order approximation, providing insight into the evolution of the wave packet envelope. This endeavor promises a deeper understanding of the interplay between electromagnetic waves and bi-isotropic nonlinear mediums, paving the way for potential applications in optics and microwaves, exemplified by our case study on chiroptic fibers. Within this section, our focus lies on the analysis and modeling of light pulse propagation within a bi- isotropic fiber, leveraging our newly formulated constitutive equations [22]. Hence, we have employed the extended 𝐺’/𝐺 -expansion method as our chosen approach for solving the nonlinear Schrödinger equation, facilitating the acquisition of precise solutions for bi-isotropic fiber nonlinearities. As delineated in our formalism expounded in [19-23], the constitutive equations governing the bi- anisotropic nonlinear effects are delineated as follows: D⃗⃗ = ε̅ E⃗⃗ + ζ̅𝐸𝐻 H⃗⃗ (9) B⃗⃗ = μ̅H⃗⃗ + ζ̅HE g E⃗⃗ (10) The medium effects are contained in the dyadic: ε̅ , μ̅, ζ̅𝐸𝐻 and ζ̅HE g due to anisotropy. The bi-isotropic medium has a Kerr type nonlinearity characterized by [19- 23]: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 576 https://internationalpubls.com εg = ε + εKerr|E 2| (11) ζHE g = ζEH ∗ + ζHE Kerr|E2| (12) ζEH ∗ is the linear bi-isotropy coefficient , and the term ζHE Kerr|E2|corrects the bi-isotropic coefficient with a quantity proportional to the field intensity [19-23]. The linear bi-isotropy factors are written as follows [19-23]: ζEH = γ − jκ (13) γ is the non-reciprocity parameter, and κ is the linear chirality parameter. γKerr is the nonlinear non-reciprocity parameter, and κKerr is the nonlinear chirality parameter. There exist three distinct cases for the biisotropic medium: 1. The chiral medium, which is reciprocal (purely imaginary), denoted by κ≠0 and γ=0. 2. The Tellegen medium, which is non-reciprocal (purely real), indicated by κ=0 and γ≠0. 3. The biisotropic medium, characterized by both chirality and non-reciprocity (complex numbers), with κ≠0 and γ≠0. In this study, our focus centers on the third case. A biisotropic fiber refers to an optical fiber featuring a chiral core enveloped by an optical cladding. The core of the biisotropic fiber possesses a slightly higher refractive index compared to the sheath. This variation in refractive index induces total internal reflection of light within the chiral core, enabling the propagation of light with two distinct modes: a right circular polarized wave (RCP) and a left circular polarized wave (LCP), each exhibiting different wave vectors. From Maxwell's equations, which serve as the cornerstone of electromagnetism and locally describe the evolution and properties of electric and magnetic fields, we specifically consider Maxwell's first equation, known as the Maxwell-Faraday equation. This equation elucidates the phenomenon of electromagnetic induction first discovered by Faraday: ∇⃗⃗ × E⃗⃗ = − ∂B⃗⃗ ∂t (14) As for the second equation, which is the Maxwell-Ampere equation, and which stems from Ampère's theorem, it links the evolution of the electric field as a function of the magnetic field. It is given by: ∇⃗⃗ × H⃗⃗ = ∂D⃗⃗ ∂t (15) What allowed us to deduce the equation of propagation in a Kerr-biisotropic medium our result is also a generalization: ∇⃗⃗ 2. E⃗⃗ − (με − μ0ε0|ζEH 2 |) ∂2E⃗⃗ ∂t2 −√μ0ε0(ζEH ∗ − ζEH) ∂∇⃗⃗ ×E⃗⃗ ∂t = (μεKerr − μ0ε0ζEHζHE Kerr)|E⃗⃗ | 2 ∂2E⃗⃗ ∂t2 + √μ0ε0ζHE Kerr|E⃗⃗ | 2 ∂∇⃗⃗ ×E⃗⃗ ∂t + μσ ∂E⃗⃗ ∂t (16) σ is the absorption coefficient. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 577 https://internationalpubls.com The electric field in the bi-isotropic fiber can be represented by wave propagating in the z direction: E±⃗⃗ ⃗⃗ = (ex⃗⃗ ⃗ ± iey⃗⃗ ⃗)Ψ±(r, t)e −i(k±z−ω0t) (17) E±⃗⃗ ⃗⃗ = Ψ±⃗⃗ ⃗⃗ ⃗(r, t)e −i(k±z−ω0t) (18) where the wave numbers k+ (RCP) and k− (LCP) can be written as: k+ = k√μ0ε0 +√με − γ2μ0ε0 (19) k− = −k√μ0ε0 +√με − γ2μ0ε0 (20) The conditions of slowly variant envelope are given by [12, 13]: | ∂2 ∂z2 Ψ±| ≪ |2ik±| ; | ∂ ∂z Ψ±| ≪ |iω0Ψ±| (21) | ∂2 ∂z2 |Ψ±| 2 Ψ±| ≪ |iω0 ∂ ∂t |Ψ±| 2 Ψ±| ≪ |iω0|Ψ±| 2 Ψ±| (22) A±(z,t) 2k± ⋅ ez⃗⃗ ⃗ = Ψ±⃗⃗ ⃗⃗ ⃗ (23) The phenomenon of dispersion is included in heuristic form through the relation Δk = 1 v (24) After algebraic manipulations, within the slowly varying amplitude approximation of Maxwell’s equations [18-23], signals’ propagating through bi-isotropic fiber is described using the following equation: ( ∂ ∂z A±(z, t) + (β1 ∂ ∂t + i 1 2 β2 ∂2 ∂t2 − 1 6 β3 ∂3 ∂t3 ) A±(z, t)) = iδA±(z, t)iρ|A±(z, t)| 2 A±(z, t) (25) Where the chromatic dispersion coefficients associated with β1, β2 and β3, and α is the attenuation coefficient and fiber nonlinearity related to coefficient ρ .When setting the variable t̃ = t − β1z, we obtain the nonlinear Schrödinger equation as follows: ∂ ∂z A±(z, t̃) + (i 1 2 β2 ∂2 ∂t̃2 − 1 6 β3 ∂3 ∂t̃3 ) A±(z, t̃) = ( −α 2 + iδ) A±(z, t̃) − iρ|A±(z, t̃)| 2 A±(z, t̃) (26) In this step the bi-isotropic fiber is operattin in the third optical windows, where β3 = 0 and neglecting absorption α = 0 the Eq.(26) becomes ∂ ∂z A±(z, t̃) + (i 1 2 β2 ∂2 ∂t̃2 ) A±(z, t̃) = iδA±(z, t̃) − iρ|A±(z, t̃)| 2 A±(z, t̃) (27) Since A± = A±(z, t̃) is a complex function , we assume that travelling wave transformation is in the form A±(z, t̃) = v±(z, t̃)exp(iθ±(z, t̃)) (28) where v±(z, t̃) and θ±(z, t̃) are amplitude and phase functions respectively.Substituting the wave transformation Eq.(17) into Eq.(18) and separating the real and imaginary parts , we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 578 https://internationalpubls.com −v±θ±z − 1 2 β2(z)(v±t̃t̃ − v±θ±z 2) − ρ±v± 3 = 0 (29) v±z − β2(z)(2θ±zv±z + v±θ±zz) − v± = 0 (30) where θ±z = dθ± dz , θ±zz = d2θ± dz2 and v±z = dv± dz , v±t̃ = dv± dt̃ Considering the homogenous balance in Eq.(29) and Eq.(30), we assume that Eq.(29) and Eq.(30) have the following solutions form { v±(z, t̃) = v±(ξ) = a0 + a1 ( G± ′ G± ) + b1√1 + 1 μ ( G± ′ G± ) 2 ξ± = p±(z)t̃ + q±(z) (31) θ±(z, t̃) = a±(z)t̃ 2 + b±(z)t̃ + c±(z) (32) where G± = G±(ξ) satifies { −a0a± ′(z) + 2a0a± 2(z)β±2(z) = 0 −a1a± ′(z) + 2a1a± 2(z)β±2(z) = 0 −b1a± ′(z) + 2b1a± 2(z)β±2(z) = 0 b1b± ′(z) + 2b1a± 2(z)b±(z)β±2(z) = 0 −a1β±2(z) − a1 3ρ± − 3a1b1 2ρ± μ = 0 3a0a1 2 + 3a0b1 2 μ = 0 − 1 2 μb1β±2(z)p 2(z) − ρ±b1 3 − b1c± ′(z) − 3a0b1ρ± + 1 2 b1β±2(z)b 2 = 0 −a1c± ′(z) − a1μβ±2(z)p± 2(z) − 3ρ±a0 2a1 − 3a1b1 2ρ± + 1 2 a1b 2β±2(z) = 0 p± ′(z) − 2p±(z)ρ±β±2(z) q± ′(z) + p±(z)β±2(z)b = 0 −ρ±β±2(z) = 1 (33) Solving the algebric system with the help of Mathematica , we get the following cases. Case 1 a0 = 0, b1 = 0, a1 = a1, p±(z) = c2a±(z), q±(z) = 1 2 c1c2a±(z) + c3 (34) b±(z) = c1a±(z), c±(z) = 1 4 (c1 2 − 2μc2 2)a±(z) + c4 (35) ρ±(z) = c2 2β±(z)a± 2(z) a1 2 (36) where ci(i = 1,2, … ,5) are arbitry constants and a±(z) is given by a±(z)[−2∫ β±2(z)dz + c5] = 1 (37) The exact solution of Eq.(26) is given by, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 579 https://internationalpubls.com A±(z, t̃) = { √−μa1 ( A1sinh√−μξ+A2cosh√−μξ A1cosh√−μξ+A2sinh√−μξ ) . exp(iθ±(z, t̃)); μ < 0 √μa1 ( A1sin√μξ+A2cos√μξ A2sin√μξ−A1cos√μξ ) . exp(iθ±(z, t̃)); μ > 0 A2 A1+A2 ; μ = 0 (38) The dark solitary of Eq.(27) are given in the form A±1(z, t̃) = √−μa1tanh(√−μξ + ξ0) ⋅ exp{ia±(z, t)[t̃ 2 + c1t̃ + (c1 2 − 2μc2 2]} (39) The hyperbolic function solution of Eq.(27) are written as A±2(z, t̃) = √−μa1coth(√−μξ + ξ0) ⋅ exp{ia±(z, t)[t̃ 2 + c1t̃ + (c1 2 − 2μc2 2]} (40) And the triangular periodic wave solution of Eq.(27) are written as A±3(z, t̃) = √−μa1cot(√−μξ + ξ0) ⋅ exp{ia±(z, t)[t̃ 2 + c1t̃ + (c1 2 − 2μc2 2]} (41) Fig.1. Numerical simulation of RCP Amplitude A+1(z, t̃) for the first case of solutions where: c1=3 , c2=4 , c3=5 ξ0 =1; μ=-1/2 , a1 =3 , b1=3, μr = 1.4, εr = 5.2, γ = 0.5, κ = 0.5,γKerr = 2.58 × 10−17 κKerr = 3.58 × 10−17. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 580 https://internationalpubls.com Fig.2. Numerical simulation of LCP Amplitude A−1(z, t̃) for the first case of solutions where: c1=3 , c2=4 , c3=5 ξ0 =1; μ=-1/2 , a1 =3 , b1=3, μr = 1.4, εr = 5.2, γ = 0.5, κ = 0.5,γKerr = 2.58 × 10−17 κKerr = 3.58 × 10−17. Case 2 a0 = 0; a1 = 0, b1 = b1, p±(z) = c2a(z) , b±(z) = c1a±(z) (42) q±(z) = 1 2 c1c2a±(z) + c3 ; −a±(z)β±(z) = 1 (43) c±(z) = 1 4 (c1 2 + μc2 2)a±(z) + c4 (44) a±(z)[−2∫ β±2(z)dz + c5] = 1 (45) In this case , we have the exact solution of Eq.(27) as follows A±(z, t̃) = { b1√1 − ( A1sinh√−μξ+A2cosh√−μξ A1cosh√−μξ+A2sinh√−μξ ) 2 . exp(iθ±(z, t̃)); μ < 0 b1√1 + ( a1sin√μξ+a2cos√μξ a2sin√μξ−a1cos√μξ ) 2 . exp(iθ±(z, t̃)); μ > 0 (46) The bright solitary wave solution of Eq.(27) are given by A±4(z, t̃) = b1sec(√−μξ + ξ0). exp{ia±(z, t̃)[t̃ 2 + c1t̃ + (c1 2 − 2μc2 2]} (47) The singular solution of Eq.(27) are given by A±5(z, t̃) = b1csch(√−μξ + ξ0). exp{ia±(z, t̃)[t̃ 2 + c1t̃ + (c1 2 − 2μc2 2]} (48) The triangular periodic solution of Eq.(27) are given by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 581 https://internationalpubls.com A±6(z, t̃) = b1csch(√−μξ + ξ0) ⋅ exp{ia±(z, t̃)[t̃ 2 + c1t̃ + (c1 2 − 2μc2 2]} (49) Fig.3. Numerical simulation of RCP Amplitude A+4(z, t̃) for the first case of solutions where: c1=3 , c2=4 , c3=5 ξ0 =1; μ=-1/2 , a1 =3 , b1=3, μr = 1.4, εr = 5.2, γ = 0.5, κ = 0.5,γKerr = 2.58 × 10−17 κKerr = 3.58 × 10−17. . Fig.4. Numerical simulation of LCP Amplitude A−4(z, t̃) for the first case of solutions where: c1=3 , c2=4 , c3=5 ξ0 =1; μ=-1/2 , a1 =3 , b1=3, μr = 1.4, εr = 5.2, γ = 0.5, κ = 0.5,γKerr = 2.58 × 10−17 κKerr = 3.58 × 10−17. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 582 https://internationalpubls.com Case 3 a0 = 0, a1 = a1, b1 = b1, p±(z) = c2a±(z) (50) q±(z) = 1 2 c1c2a±(z) + c3; b±(z) = c1a±(z) (51) c±(z) = 1 8 (c1 2 − 2μc2 2)a±(z) + c1 (52) −a±(z)β±2(z) = 1 ; ρ±(z) = c1 2β±2(z)a± 2(z) 4b1 2 (53) Where a±(z) satisfies the following condition a±(z)[−2∫ β±2(z)dz + c5] = 1 (54) In this case we have to type of exact solutions for the Eq.(27) • μ < 0 , the hyperbolic solutions are defined by A±7(z, t̃) = √−μa1 [ A1sinh√−μξ+A2cosh√−μξ A1cosh√−μξ+A2sinh√−μξ ∓ i√( A1sinh√−μξ+A2cosh√−μξ A1cosh√−μξ+A2sinh√−μξ ) 2] . exp[iθ±(z, t̃)] (55) • If μ < 0, cothξ0 = A2 A1 , | A1 A2 | > 1,then the bright-dark soliton of Eq.(27) are derived as, A±7,1(z, t̃) = √−μa1[tanh√−μξ + ξ0] = ∓isec(√−μξ + ξ0). exp[iθ±(z, t̃)] (56) • If μ < 0, cothξ0 = A2 A1 , | A1 A2 | < 1,then the solution of Eq.(27) becomes A±7,2(z, t̃) = √−μa1[coth√−μξ + ξ0] = ±csch(√−μξ + ξ0). exp[iθ±(z, t̃)] (57) • μ > 0 the triangular periodic sulution are given by A±8(z, t̃) = √−μa1 [ A1cos√−μξ+A2sin√−μξ A1cos√−μξ+A2sin√−μξ ∓ i√( A1cos√−μξ+A2sin√−μξ A1cos√−μξ+A2sin√−μξ ) 2] . exp[iθ±(z, t̃)] (58) • If μ > 0 ,tanξ0 = A2 A1 , hen the solution of Eq.(27) becomes as follow A±8,1(z, t̃) = √μa1[cot√μξ + ξ0] = ±csch(√μξ + ξ0). exp[iθ±(z, t̃)] (59) Where θ±(z, t̃) = a±(z) [t̃ 2 + c1t̃ + c1 2−2μc2 2 8 a±(z) + c4] (60) ξ± = c2a±(z)t̃ + 1 2 c1c2a±(z) + c3 (61) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 583 https://internationalpubls.com Fig.5. Numerical simulation of RCP Amplitude A+7(z, t̃) for the first case of solutions where: c1=3 , c2=4 , c3=5 ξ0 =1; μ=-1/2 , a1 =3 , b1=3, μr = 1.4, εr = 5.2, γ = 0.5, κ = 0.5,γKerr = 2.58 × 10−17 κKerr = 3.58 × 10−17. Fig.5. Numerical simulation of LCP Amplitude A−7(z, t̃) for the first case of solutions where: c1=3 , c2=4 , c3=5 ξ0 =1; μ=-1/2 , a1 =3 , b1=3, μr = 1.4, εr = 5.2, γ = 0.5, κ = 0.5,γKerr = 2.58 × 10−17 κKerr = 3.58 × 10−17. In the following figures 1-6, the different exact families of solutions were obtained and we take as an example some graphs for the solutions of case 1, case 2, and case 3 with two modes of Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 584 https://internationalpubls.com propagation (RCP) and (LCP). The extended G'/G-expansion method has been used to present an analytic study of the generalized nonlinear Schrödinger equation in bi-isotropic (chiral and non- reciprocal). optical fibers. The method is constructed on a perturbation expansion in powers of a dimensionless parameter, which allows for the inclusion of varying dispersion and nonlinearity. This types the method well-suited for modeling a wide range of optical fibres, including those with complex dispersion and nonlinearity profiles. The method was used to find exact different families of solutions of the nonlinear Schrödinger equation for a bi-isotropic optical fiber. The method was able to accurately describe the propagation of optical pulses in bi-isotropic (chiral and non-reciprocal). fibers and can be used to predict the behavior of these pulses under different conditions. The method can also be used to study the dynamics of solitons, including the stability and interactions of these localized structures. 5. Results This investigate is concerned with a new formulation of constitutive relation linking to the magnetic effect, to understand rigorously the physical nature of biisotropic effects and to generalize the main macroscopic models. We inferred the nonlinear Schrodinger equation for a bi-isotropic medium term with a nonlinear term of magnetizing. In this article, the extended (G'/G)-expansion method is a powerful technique for determining a family of solutions of the nonlinear Schrödinger equation in bi- isotropic optical fibers. This method is based on the use of a perturbation expansion in powers of the dimensionless parameter, and it is valid for both weak and strong nonlinearities. The method allows for the inclusion of varying dispersion and nonlinearity, making it well-suited for modeling a wide range of optical fibres. Overall, the extended (G'/G)-expansion method is a valuable tool for understanding the dynamics of nonlinear optical systems, and it is expected to have a wide range of applications in the field of nonlinear optics. Refrences [1] Gao, X.: Variety of the cosmic plasmas: general variable-coefficient Korteweg-de Vries–Burgers equation with experimental/observational support. Europhys. Lett.110, 15002 (2015). [2] Zhen, H.-L., Tian, B., Wang, Y.-F.: Soliton solution and chaotic motions of the Zakharov equation for the Langmuir wave in the plasma. Phys. Plasmas 22, 1–9 (2015). [3] Sun, W.-R., Tian, B., Jiang, Y. Optical rogue waves associated with the negative coherent coupling in an isotropic medium. Phys. Rev. E 91, 023205 (2015). [4] Wang, Y.-F., Tian, B., Wang, M. Solitons via an auxiliary function for an inhomogeneous higher-order nonlinear Schrödinger equation in optical fiber communications. Nonlinear Dyn. 79, 721–729 (2015). [5] Gao, X.-Y.Bäcklund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variablecoefficient B-type Kadomtsev-Petviashvili equation in fluid mechanics. Ocean Eng. 96, 245–247 (2015). [6] ao, X.-Y. Incompressible-fluid symbolic computation and bäcklund transformation: (3 + 1)-dimensional variablecoefficient boiti-leon-manna-pempinelli model. Z. Naturforsch. A. 70, 59–61 (2015). [7] Biswas, A., Mirzazadeh, M., Savescu, M., Milovic, D, Khan, K.R., Mahmood, M.F., Belic, M.: Singular solitons in optical metamaterials by ansatz method and simplest equation approach. J. Mod. Opt. 61, 1550–1555 (2014). [8] Mirzazadeh, M. Arnous, A.H., Mahmood, M.F., Zerrad, E., Biswas, A.: Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coefficients by trial solution approach. Nonlinear Dyn. 81(1–2), 277- 282 (2015). [9] Mirzazadeh, M., Zhou, Q., Bhrawy, A.H., Biswas, A., Belic,M.: Optical solitons in cascaded system with three integration schemes. J. Optoelectron. Adv. Mater. 17(1–2), 74–81 (2015). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 585 https://internationalpubls.com [10] Eslami, M., Mirzazadeh, M., Vajargah, B.F., Biswas, A.:Optical solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients by the first integralmethod. OptikInt. J. Light Electron Opt. 125(13), 3107–3116 (2014). [11] Mirzazadeh, M., Biswas, A.: Optical solitons with spatiotemporal dispersion by first integral approach and functional variable method. Optik Int. J. Light Electron Opt. 125(19), 5467–5475 (2014). [12] Mirzazadeh, M., Zhou, Q, Zerrad, E., Mahmood, M.F.,Biswas, A., Belic,M.: Bifurcation analysis and bright soliton of generalized resonant dispersive nonlinear Schrodinger’s equation. Optoelectron. Adv. Mater. 9(11–12), 1342–1346 (2015). [13] Mirzazadeh, M, Mahmood, M.F., Majid, F.B., Biswas, A.,Belic, M.: Optical solitons in birefringent fibers with Riccati equation method. Optoelectron. Adv. Mater. 7–8, 1032–1036 (2015). [14] Wang M, Li X, Zhang J. The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys Lett A. 2008;372: 417–423. [15] Zayed EM, Alurrfi KA. On solving two higher-order nonlinear PDEs describing the propagation of optical pulses in optic fibers using the ((G′/G),(1/G))-expansion method. Ricerche Mat. 2015;64:164–194. [16] Zayed EM, Shahoot AM, Alurrfi KA. The ((G′/G),(1/G))- expansion method and its applications for constructing many new exact solutions of the higher-order nonlinear Schrödinger equation and the quantum Zakharov - Kuznetsov equation. Opt. Quant. Electron. 2018;50.doi:10.1007/s11082-018-1337. [17] MALIK, Sandeep, KUMAR, Sachin, and NISAR, Kottakkaran Sooppy. Invariant soliton solutions for the coupled nonlinear Schrödinger type equation. Alexandria Engineering Journal, 2023, vol. 66, p. 97-105. [18] Yan ZY. Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres. Chaos Solitons Fractals. 2003;16: 759–766. [19] Z. Mezache and F. Benabdelaziz, Study of chiroptical fiber nonlinearities with new formulation of constitutive equations, J. Electromagn. Waves Appl. 29 (2015), 2257–2268. [20] Z. Mezache and F. Benabdelaziz, Rigorous approach of the constitutive relations for nonlinear chiral media, Prog. Electromagn. Res. Lett. 52 (2015), 57–62. [21] Mezache, S. Aib, F. Benabdelaziz and C. Zebiri, Modeling of a light pulse in bi-isotropic optical fiber with Kerr effect: Case of Tellegen media, Nonlinear Dyn. 86 (2) (2016), 789–794. [22] Zinelabiddine, Mezache, and Benabdelaziz Fatiha. "Effect of the Nonlinear Parameters on the Propagation in Bi- isotropic Media." International Journal of Nonlinear Sciences and Numerical Simulation 18.6 (2017): 541-547. [23] Mezache, Zinelabiddine, and Fatiha Benabdelaziz. "Nonlinear effects in chiral nihility metamaterial." Optical and Quantum Electronics 50.8 (2018): 1-10. [24] AMEEN, Ismail Gad, TAIE, Rasha Osman Ahmed, and ALI, Hegagi Mohamed. Two effective methods for solving nonlinear coupled time-fractional Schrödinger equations. Alexandria Engineering Journal, 2023, vol. 70, p. 331-347.