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Arab Journal of Basic and Applied Sciences Volume 31, 2024 - Issue 1

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Research Article

Extraction of optical wave structures to the coupled fractional system in magneto-optic waveguides

Jan Muhammada Department of Mathematics, Shanghai University, Shanghai, China

Muhammad Bilal Riazb IT4Innovations, VSB – Technical University of Ostrava, Ostrava, Czech Republic;c Department of Computer Science and Mathematics, Lebanese American University, Byblos, LebanonCorrespondence[email protected]

Usman Younasa Department of Mathematics, Shanghai University, Shanghai, China

Naila Nasreend Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, People’s Republic of China

Adil Jhangeerb IT4Innovations, VSB – Technical University of Ostrava, Ostrava, Czech Republic;e Department of Mathematics, Namal University, Mainwali, Pakistan

Dianchen Lud Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, People’s Republic of China

Pages 242-254 | Received 24 Jan 2024, Accepted 28 Mar 2024, Published online: 10 Apr 2024

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https://doi.org/10.1080/25765299.2024.2337469
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Figure 1. Sketches of the EquationEq. (20)(20) $\begin{array}{l} p_{1} (x, t) = - \frac{3 exp (\frac{i Γ (β + 1) (\frac{t^{ϵ} (κ^{2} (a_{1} + ζ a_{2}) + f_{2} - ζ f_{1} + κ (ζ σ_{2} - σ_{1}))}{1 - \frac{Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ}} + φ - κ x^{ϵ})}{ϵ})}{Γ_{2}} \\ \times (φ_{1} - i \sqrt{2} \sqrt{φ_{1}} \sqrt{- \frac{φ_{1}}{φ_{2}}} \sqrt{φ_{2}} sech (\sqrt{φ_{1}} (\frac{Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ} + ϑ))) . \end{array}$ (20) for $ϵ = 0.6 .$

Figure 1. Sketches of the EquationEq. (20)(20) p1(x,t)=−3 exp (iΓ(β+1)(tϵ(κ2(a1+ζa2)+f2− ζf1+κ(ζσ2−σ1))1−Γ(β+1)(xϵ−ρtϵ)ϵ+φ−κxϵ)ϵ)Γ2×(φ1−i2φ1−φ1φ2φ2sech(φ1(Γ(β+1)(xϵ−ρtϵ)ϵ+ϑ))).(20) for ϵ=0.6.

Figure 2. Sketches of the EquationEq. (22)(22) $\begin{array}{l} p_{3} (x, t) = \frac{3 exp (\frac{i Γ (β + 1) (\frac{ϵ t^{ϵ} (κ (κ (a_{1} + ζ a_{2}) + ζ σ_{2} - σ_{1}) + f_{2} - ζ f_{1})}{Γ (β + 1) (ρ t^{ϵ} - x^{ϵ}) + ϵ} + φ - κ x^{ϵ})}{ϵ})}{Γ_{2}} \\ \times (- 1 + \frac{2 i \sqrt{2} \sqrt{β_{1} β_{2}} e^{\sqrt{φ_{1}} (\frac{Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ} + ϑ)}}{β_{1} e^{2 \sqrt{φ_{1}} (\frac{Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ} + ϑ)} - β_{2}}) φ_{1} . \end{array}$ (22) for $ϵ = 0.2 .$

Figure 2. Sketches of the EquationEq. (22)(22) p3(x,t)=3 exp (iΓ(β+1)(ϵtϵ(κ(κ(a1+ζa2)+ζσ2−σ1)+f2−ζf1)Γ(β+1)(ρtϵ−xϵ)+ϵ+φ−κxϵ)ϵ)Γ2×(−1+2i2β1β2eφ1(Γ(β+1)(xϵ−ρtϵ)ϵ+ϑ)β1e2φ1(Γ(β+1)(xϵ−ρtϵ)ϵ+ϑ)−β2)φ1.(22) for ϵ=0.2.

Figure 3. Sketches of the EquationEq. (25)(25) $\begin{array}{l} p_{6} (x, t) = - \frac{3 exp (\frac{i Γ (β + 1) (\frac{ϵ t^{ϵ} (a_{1} κ^{2} + f_{2} + ζ a_{2} κ^{2} - ζ f_{1} + ζ σ_{2} κ - κ σ_{1})}{Γ (β + 1) (ρ t^{ϵ} - x^{ϵ}) + ϵ} + φ - κ x^{ϵ})}{ϵ})}{Γ_{2}} \\ \times \sqrt{φ_{1}} (\sqrt{φ_{1}} - \sqrt{- \frac{φ_{1}}{φ_{2}}} \sqrt{φ_{2}} (sech (\sqrt{2} \sqrt{- φ_{1}} (\frac{Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ} + ϑ)) \\ + i tanh (\sqrt{2} \sqrt{- φ_{1}} (\frac{Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ} + ϑ)))) . \end{array}$ (25) for $ϵ = 0.67 .$

Figure 3. Sketches of the EquationEq. (25)(25) p6(x,t)=−3 exp (iΓ(β+1)(ϵtϵ(a1κ2+f2+ζa2κ2−ζf1+ζσ2κ−κσ1)Γ(β+1)(ρtϵ−xϵ)+ϵ+φ−κxϵ)ϵ)Γ2×φ1(φ1−−φ1φ2φ2(sech(2−φ1(Γ(β+1)(xϵ−ρtϵ)ϵ+ϑ))+itanh(2−φ1(Γ(β+1)(xϵ−ρtϵ)ϵ+ϑ)))).(25) for ϵ=0.67.

Figure 4. Sketches of the EquationEq. (29)(29) $\begin{array}{l} p_{10} (x, t) = \frac{exp (\frac{i Γ (β + 1) (\frac{ϵ t^{ϵ} (κ (κ (a_{1} + ζ a_{2}) + ζ σ_{2} - σ_{1}) + f_{2} - ζ f_{1})}{Γ (β + 1) (ρ t^{ϵ} - x^{ϵ}) + ϵ} + φ - κ x^{ϵ})}{ϵ})}{\sqrt{Γ_{1}}} \\ (\sqrt{2} \sqrt{- \frac{φ_{1}}{φ_{2}}} \sqrt{φ_{2}} csc (\sqrt{- φ_{1}} (\frac{Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ} + ϑ)) - i \sqrt{φ_{1}}) . \end{array}$ (29) for $ϵ = 0.545 .$

Figure 4. Sketches of the EquationEq. (29)(29) p10(x,t)= exp (iΓ(β+1)(ϵtϵ(κ(κ(a1+ζa2)+ζσ2−σ1)+f2−ζf1)Γ(β+1)(ρtϵ−xϵ)+ϵ+φ−κxϵ)ϵ)Γ1(2−φ1φ2φ2csc(−φ1(Γ(β+1)(xϵ−ρtϵ)ϵ+ϑ))−iφ1).(29) for ϵ=0.545.

Figure 5. Sketches of the EquationEq. (33)(33) $\begin{array}{l} p_{14} (x, t) = \frac{\sqrt{φ_{1}} exp (\frac{i Γ (β + 1) (\frac{ϵ t^{ϵ} (κ (κ (a_{1} + ζ a_{2}) + ζ σ_{2} - σ_{1}) + f_{2} - ζ f_{1})}{Γ (β + 1) (ρ t^{ϵ} - x^{ϵ}) + ϵ} + φ - κ x^{ϵ})}{ϵ})}{\sqrt{Γ_{1}}} \\ \times (\frac{4 \sqrt{2} \sqrt{φ_{1}} \sqrt{φ_{2}} e^{\sqrt{φ_{1}} (\frac{Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ} + ϑ)}}{4 φ_{1} φ_{2} e^{2 \sqrt{φ_{1}} (\frac{Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ} + ϑ)} - 1} - i) . \end{array}$ (33) for $ϵ = 0.45 .$

Figure 5. Sketches of the EquationEq. (33)(33) p14(x,t)=φ1 exp (iΓ(β+1)(ϵtϵ(κ(κ(a1+ζa2)+ζσ2−σ1)+f2−ζf1)Γ(β+1)(ρtϵ−xϵ)+ϵ+φ−κxϵ)ϵ)Γ1 ×(42φ1φ2eφ1(Γ(β+1)(xϵ−ρtϵ)ϵ+ϑ)4φ1φ2e2φ1(Γ(β+1)(xϵ−ρtϵ)ϵ+ϑ)−1−i).(33) for ϵ=0.45.

Figure 6. Sketches of the EquationEq. (37)(37) $\begin{array}{l} p_{2} (x, t) = \frac{6 exp (\frac{i Γ (β + 1) (\frac{ϵ t^{ϵ} (\frac{{(σ_{1} - σ_{2})}^{2} (a_{1} + ζ a_{2})}{4 {(a_{1} - a_{2})}^{2}} + \frac{(σ_{2} - σ_{1}) (ζ σ_{2} - σ_{1})}{2 (a_{1} - a_{2})} + f_{2} - ζ f_{1})}{Γ (β + 1) (ρ t^{ϵ} - x^{ϵ}) + ϵ} + \frac{(σ_{1} - σ_{2}) x^{ϵ}}{2 (a_{1} - a_{2})} + φ)}{ϵ})}{Γ_{2}} \\ \times (\sqrt{- δ_{0}} - i \sqrt{δ_{0}} tanh (\frac{\sqrt{- δ_{0}} Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ}) \sqrt{- δ_{0}}) . \end{array}$ (37) for $ϵ = 0.29 .$

Figure 6. Sketches of the EquationEq. (37)(37) p2(x,t)=6 exp (iΓ(β+1)(ϵtϵ((σ1−σ2)2(a1+ζa2)4(a1−a2)2+(σ2−σ1)(ζσ2−σ1)2(a1−a2)+f2−ζf1)Γ(β+1)(ρtϵ−xϵ)+ϵ+(σ1−σ2)xϵ2(a1−a2)+φ)ϵ)Γ2×(−δ0−iδ0 tanh(−δ0Γ(β+1)(xϵ−ρtϵ)ϵ)−δ0).(37) for ϵ=0.29.

Figure 7. Sketches of the EquationEq. (40)(40) $\begin{array}{l} p_{5} (x, t) = \frac{6 exp (\frac{i Γ (β + 1) (\frac{ϵ t^{ϵ} (\frac{{(σ_{1} - σ_{2})}^{2} (a_{1} + ζ a_{2})}{4 {(a_{1} - a_{2})}^{2}} + \frac{(σ_{2} - σ_{1}) (ζ σ_{2} - σ_{1})}{2 (a_{1} - a_{2})} + f_{2} - ζ f_{1})}{Γ (β + 1) (ρ t^{ϵ} - x^{ϵ}) + ϵ} + \frac{(σ_{1} - σ_{2}) x^{ϵ}}{2 (a_{1} - a_{2})} + φ)}{ϵ})}{Γ_{2}} \\ \times (- 1 + \frac{i \sqrt{δ_{0}} (4 sinh (\frac{2 \sqrt{- δ_{0}} Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ}) + 3)}{\sqrt{- δ_{0}} (4 cosh (\frac{2 \sqrt{- δ_{0}} Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ}) - 5)}) δ_{0} . \end{array}$ (40) for $ϵ = 0.331 .$

Figure 7. Sketches of the EquationEq. (40)(40) p5(x,t)=6 exp (iΓ(β+1)(ϵtϵ((σ1−σ2)2(a1+ζa2)4(a1−a2)2+(σ2−σ1)(ζσ2−σ1)2(a1−a2)+f2−ζf1)Γ(β+1)(ρtϵ−xϵ)+ϵ+(σ1−σ2)xϵ2(a1−a2)+φ)ϵ)Γ2×(−1+iδ0(4sinh(2−δ0Γ(β+1)(xϵ−ρtϵ)ϵ)+3)−δ0(4cosh(2−δ0Γ(β+1)(xϵ−ρtϵ)ϵ)−5))δ0.(40) for ϵ=0.331.

Figure 8. Sketches of the EquationEq. (41)(41) $\begin{array}{l} p_{6} (x, t) = \frac{6 exp (\frac{i Γ (β + 1) (\frac{ϵ t^{ϵ} (\frac{{(σ_{1} - σ_{2})}^{2} (a_{1} + ζ a_{2})}{4 {(a_{1} - a_{2})}^{2}} + \frac{(σ_{2} - σ_{1}) (ζ σ_{2} - σ_{1})}{2 (a_{1} - a_{2})} + f_{2} - ζ f_{1})}{Γ (β + 1) (ρ t^{ϵ} - x^{ϵ}) + ϵ} + \frac{(σ_{1} - σ_{2}) x^{ϵ}}{2 (a_{1} - a_{2})} + φ)}{ϵ})}{Γ_{2}} \\ \times (- 1 + \frac{i \sqrt{δ_{0}} (c sinh (\frac{2 \sqrt{- δ_{0}} Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ}) + d)}{4 c \sqrt{- δ_{0}} cosh (\frac{2 \sqrt{- δ_{0}} Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ}) - λ \sqrt{δ_{0} (- (c^{2} + d^{2}))}}) δ_{0} . \end{array}$ (41) for $ϵ = .41 .$

Figure 8. Sketches of the EquationEq. (41)(41) p6(x,t)=6 exp (iΓ(β+1)(ϵtϵ((σ1−σ2)2(a1+ζa2)4(a1−a2)2+(σ2−σ1)(ζσ2−σ1)2(a1−a2)+f2−ζf1)Γ(β+1)(ρtϵ−xϵ)+ϵ+(σ1−σ2)xϵ2(a1−a2)+φ)ϵ)Γ2 ×(−1+iδ0(csinh(2−δ0Γ(β+1)(xϵ−ρtϵ)ϵ)+d)4c−δ0cosh(2−δ0Γ(β+1)(xϵ−ρtϵ)ϵ)−λδ0(−(c2+d2)))δ0.(41) for ϵ=.41.

Figure 9. Sketches of the EquationEq. (44)(44) $\begin{array}{l} p_{9} (x, t) = \frac{1}{2} exp (\frac{i Γ (β + 1) (\frac{(σ_{1} - σ_{2}) x^{ϵ}}{2 (a_{1} - a_{2})} + ω t^{ϵ} + φ)}{ϵ}) \\ \times (2 + \frac{\sqrt{2} ((1 - δ_{0}) tan (\frac{2 \sqrt{δ_{0}} Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ}) + (δ_{0} + 1) sec (\frac{2 \sqrt{δ_{0}} Γ (β + 1) (x^{ϵ} - ρ t^{ϵ})}{ϵ}))}{\sqrt{δ_{0}}}) ϕ_{0} . \end{array}$ (44) for $ϵ = .22 .$

Figure 9. Sketches of the EquationEq. (44)(44) p9(x,t)=12exp (iΓ(β+1)((σ1−σ2)xϵ2(a1−a2)+ωtϵ+φ)ϵ) ×(2+2((1−δ0) tan (2δ0Γ(β+1)(xϵ−ρtϵ)ϵ)+(δ0+1) sec (2δ0Γ(β+1)(xϵ−ρtϵ)ϵ))δ0)ϕ0.(44) for ϵ=.22.

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Extraction of optical wave structures to the coupled fractional system in magneto-optic waveguides

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Extraction of optical wave structures to the coupled fractional system in magneto-optic waveguides

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