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A new version of the motion equations of pseudo null curves with compatible Hasimoto map

Authors:
Talat Körpınar at Mus Alparslan University
Talat Körpınar
Yasin Ünlütürk at Kırklareli University
Yasin Ünlütürk
  • Kırklareli University
Zeliha Körpınar
Zeliha Körpınar
Zeliha Körpınar
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Abstract and Figures

In this manuscript, we study links between nonlinear equations of motion and equations attained by taking geometric differential description of Hasimoto map of pseudo null curves and some physical consequences. First, we find the equations of Hasimoto transformation occurred by adapting Hasimoto map for pseudo null curves. Then, we investigate the correspondence of both Landau-Lifshitz system and vortex filament equation by means of the Hasimoto map of pseudo null curves. Moreover, we design the links of findings obtained by geometric physical issues. Finally, optical solutions are simulated by graphically physical constructions with Khater method to obtain optical solutions of fractional order differential equations.
The 3D graphics of spacetime fractional equation (26) (λ=2,ρ=1.2,θ=0.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (\lambda =2,\rho =1.2,\theta =0.5)$$\end{document}aτ1(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{1}(s,t)$$\end{document}(β=1,α=2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =2,\sigma =1)$$\end{document}, bτ2(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{2}(s,t)$$\end{document}(β=1,α=2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =2,\sigma =1)$$\end{document}, cτ3(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{3}(s,t)$$\end{document}(β=1,α=-2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =-2,\sigma =1)$$\end{document}, dτ6(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{6}(s,t)$$\end{document}(β=1,α=2,σ=2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \ (\beta =1,\alpha =2,\sigma =2)$$\end{document}
The 3D graphics of spacetime fractional equation (26) (λ=2,ρ=1.2,θ=0.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (\lambda =2,\rho =1.2,\theta =0.5)$$\end{document}aτ1(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{1}(s,t)$$\end{document}(β=1,α=2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =2,\sigma =1)$$\end{document}, bτ2(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{2}(s,t)$$\end{document}(β=1,α=2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =2,\sigma =1)$$\end{document}, cτ3(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{3}(s,t)$$\end{document}(β=1,α=-2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =-2,\sigma =1)$$\end{document}, dτ6(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{6}(s,t)$$\end{document}(β=1,α=2,σ=2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \ (\beta =1,\alpha =2,\sigma =2)$$\end{document}
… 
The 3D graphics of spacetime fractional equation (26) (λ=2,ρ=1.2,θ=0.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (\lambda =2,\rho =1.2,\theta =0.5)$$\end{document}aτ9(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{9}(s,t)$$\end{document}(β=1,α=2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =2,\sigma =1)$$\end{document}, bτ10(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{10}(s,t)$$\end{document}(β=1,α=2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =2,\sigma =1)$$\end{document}, cτ13(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{13}(s,t)$$\end{document}(β=0,α=2,σ=-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-1)$$\end{document}, dτ14(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{14}(s,t)$$\end{document}(β=0,α=2,σ=-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-1)$$\end{document}
The 3D graphics of spacetime fractional equation (26) (λ=2,ρ=1.2,θ=0.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (\lambda =2,\rho =1.2,\theta =0.5)$$\end{document}aτ9(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{9}(s,t)$$\end{document}(β=1,α=2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =2,\sigma =1)$$\end{document}, bτ10(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{10}(s,t)$$\end{document}(β=1,α=2,σ=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =1,\alpha =2,\sigma =1)$$\end{document}, cτ13(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{13}(s,t)$$\end{document}(β=0,α=2,σ=-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-1)$$\end{document}, dτ14(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{14}(s,t)$$\end{document}(β=0,α=2,σ=-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-1)$$\end{document}
… 
The 3D graphics of spacetime fractional equation (26) (λ=2,ρ=1.2,θ=0.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (\lambda =2,\rho =1.2,\theta =0.5)$$\end{document}aτ16(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{16}(s,t)$$\end{document}(β=0,α=2,σ=-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-2)$$\end{document}, bτ17(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{17}(s,t)$$\end{document}(β=0,α=2,σ=-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-2)$$\end{document},cτ18(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{18}(s,t)$$\end{document}(β=0,α=2,σ=-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-2)$$\end{document}, dτ20(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{20}(s,t)$$\end{document}(β=2,α=0,σ=3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =2,\alpha =0,\sigma =3)$$\end{document}
The 3D graphics of spacetime fractional equation (26) (λ=2,ρ=1.2,θ=0.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (\lambda =2,\rho =1.2,\theta =0.5)$$\end{document}aτ16(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{16}(s,t)$$\end{document}(β=0,α=2,σ=-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-2)$$\end{document}, bτ17(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{17}(s,t)$$\end{document}(β=0,α=2,σ=-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-2)$$\end{document},cτ18(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{18}(s,t)$$\end{document}(β=0,α=2,σ=-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =0,\alpha =2,\sigma =-2)$$\end{document}, dτ20(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{20}(s,t)$$\end{document}(β=2,α=0,σ=3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ (\beta =2,\alpha =0,\sigma =3)$$\end{document}
… 
The 2D graphics for solution τ6(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{6}(s,t)$$\end{document} of the space-time fractional Eq. (26) with distinct θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}(t=0.4,λ=2,ρ=1.2,β=1,α=2,σ=2).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t=0.4,\lambda =2,\rho =1.2,\beta =1,\alpha =2,\sigma =2).$$\end{document}
The 2D graphics for solution τ6(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{6}(s,t)$$\end{document} of the space-time fractional Eq. (26) with distinct θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}(t=0.4,λ=2,ρ=1.2,β=1,α=2,σ=2).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t=0.4,\lambda =2,\rho =1.2,\beta =1,\alpha =2,\sigma =2).$$\end{document}
… 
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Optical and Quantum Electronics (2023) 55:23
https://doi.org/10.1007/s11082-022-04225-2
A new version of the motion equations of pseudo null curves
with compatible Hasimoto map
Talat Körpınar1·Yasin Ünlütürk2·Zeliha Körpınar3
Received: 19 July 2022 / Accepted: 19 September 2022 / Published online: 15 November 2022
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022
Abstract
In this manuscript, we study links between nonlinear equations of motion and equations
attained by taking geometric differential description of Hasimoto map of pseudo null curves
and some physical consequences. First, we find the equations of Hasimoto transformation
occurred by adapting Hasimoto map for pseudo null curves. Then, we investigate the cor-
respondence of both Landau-Lifshitz system and vortex filament equation by means of the
Hasimoto map of pseudo null curves. Moreover, we design the links of findings obtained by
geometric physical issues. Finally, optical solutions are simulated by graphically physical
constructions with Khater method to obtain optical solutions of fractional order differential
equations.
Keywords Soliton equation theory ·Hasimoto map ·Landau-Lifshitz system ·Vortex
filament flow ·Conformable derivative
Mathematics Subject Classification 53A04 ·76B47 ·34A34
1 Introduction
The dynamics of Newton had paved a way for portraying moving of a particle in space-
time. Characterizing the motion of a particle has a crucial subject for applications in optics
(Goldstein et al. 2002;NcCuskey1963). The geometric analysis of a trajectory of particles in
motion has been mainly based on the curvature theory of curves in various space-time forms
(Santiago et al. 2017).
BYasin Ünlütürk
yasinunluturk@klu.edu.tr
Talat Körpınar
talatkorpinar@gmail.com
Zeliha Körpınar
zelihakorpinar@gmail.com
1Department of Mathematics, Mu¸s Alparslan University, Mu¸s, Turkey
2Department of Mathematics, Kırklareli University, Kırklareli, Turkey
3Department of Administration, Mu¸s Alparslan University, Mu¸s, Turkey
123
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In this paper, we describe \(\varvec{\nu }\)-conformable normalization and \(\varvec{\nu }\)-conformable recursional of electromagnetic fields in conformable spherical space. Then, we obtain normalized and recursional \(\varvec{\nu }\)-conformable electroosmotic \(\rho \left( \varvec{\tau }\right) ,\rho \left( \varvec{\nu }\right) ,\rho \left( \varvec{\beta }\right) \) optimistic \(\varvec{\nu }\)-conformable energy. Finally, we have optical microfluidic \(\varvec{\nu }\)-conformable normalized and recursional electroosmotic \(\rho \left( \varvec{ \tau }\right) ,\rho \left( \varvec{\nu }\right) ,\rho \left( \varvec{ \beta }\right) \) optimistic \(\varvec{\nu }\)-conformable energy in conformable spherical space.
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Nonlinear evolutions of electromagnetic flows in de Sitter 2-space
Article
  • Oct 2023
In this study, we derive novel pseudo-spherical frames in de Sitter 2-space. We obtain new connections for the nonlinear evolution of curves associated with the complex modified Korteweg–de Vries (cmKDV\(^{-}\)) equation, generalized Korteweg–de Vries (gKDV) equation, and others. We then describe some pseudo-solitonic surfaces swept out by these novel classes of nonlinearly moved curves as time evolves. Moreover, we observe the relationship between pseudo-solitonic structures and the magnetic trajectories of charged particles. Thus, we obtain a novel class of complex and non-complex evolution of the electromagnetic flows of the associated pseudo-solitonic magnetic curves. We also focus on the particular case in which the electromagnetic field of a linearly polarized light wave coupling into the optical fiber is investigated based on the inner geometric features of the novel pseudo-spherical frames in de Sitter 2-space. Finally, we investigate the corresponding energy functional to compute the Hashimoto-like transformed energy of pseudo-solitonic curves, pseudo-solitonic magnetic curves, and pseudo-solitonic electromagnetic curves. Overall, the derived results in this study could have potential applications improving the design and performance of optical fibers, developing new models for electromagnetic flows, advancing the understanding of pseudo-solitonic structures.
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Optical spherical microfluidic heisenberg spherical ferromagnetic model
Article
Full-text available
  • Oct 2023
  • OPT QUANT ELECTRON
In this manuscript, we define spherical mKdV motion conditions of electromagnetic fields. Also, we construct spherical microfluidical Heisenberg mKdV Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}\mathbf {n}$$\end{document}—electromotive forces in spherical space. Moreover, we design geometric ferromagnetic spherical microfluidical Heisenberg mKdV Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}} \mathbf {n}$$\end{document}—magnetic flux densities. Finally, we obtain spherical microfluidical Heisenberg mKdV Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}\mathbf {n}$$\end{document}—magnetic flux surfaces net in spherical space.
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On electromagnetic curves and geometric phase associated with frontals in de-Sitter 2-space
Article
  • Aug 2023
  • INDIAN J PHYS
In this work, we introduce a Berry phase model which is very important in physics relative to the frontals in de-Sitter 2-space. With this approach, we examine the relationship between a singular curve and Fermi–Walker’s law and Berry’s phase law in de-Sitter 2-space. We give the Rytov curves by using an orthonormal frame of spacelike frontals and timelike frontals with Fermi–Walker parallelism. We express parametric representations of Rytov curves associated with a singular optical fiber. We examine electromagnetic curves associated with orthonormal vectors of spacelike frontals and timelike frontals in de-Sitter 2-space. Hence, we express the magnetic field equations, Lorentz force equations and magnetic trajectory equations along a singular optical fiber. Finally, we give examples that support the theories both physically and geometrically.
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New analytical solutions for the inextensible Heisenberg ferromagnetic flow and solitonic magnetic flux surfaces in the binormal direction
Article
Full-text available
Maxwellian electromagnetism describes the wave features of the light and related subjects. Its original formulation was established 150 years ago. One of the four Maxwell's equations is Gauss's law, which states significant facts regarding magnetic flux through surfaces. It was also observed that optical media provides surface electromagnetism around 60 years ago. This observation leads to improve new techniques on nano-photonics, metamaterials, and plasmonics. The goal of this manuscript is to suggest novel accurate and local conditions for defining magnetic flux surfaces for the inextensible Heisenberg ferromagnetic flow in the binormal direction. The theoretical accuracy of the methodology is verified through the evolution of magnetic vector fields and the anti-symmetric Lorentz force field operator. On the other hand, the numerical accuracy and efficiency is developed in detail by considering the conformable fractional derivative method when these fields are transformed under the traveling wave hypothesis.
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On Optical Solitons of the Fractional (3+1)-Dimensional NLSE With Conformable Derivatives
Article
Full-text available
  • Apr 2020
In this article, the fractional (3+1)-dimensional nonlinear Shrödinger equation is analyzed with kerr law nonlinearity. The extended direct algebraic method (EDAM) is applied to obtain the optical solitons of this equation with the aid of the conformable derivative. Optical solitons are investigated for this equation with the aid of the EDAM after the nonlinear Shrödinger equation transforms an ordinary differential equation using the wave variables transformation.
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Khater method for Nonlinear Sharma Tasso-Olever (STO) Equation of Fractional Order
Article
Full-text available
  • Nov 2017
In this work, we have implemented a direct method, known as Khater method to establish exact solutions of nonlinear partial differential equations of fractional order. Number of solutions provided by this method is greater than other traditional methods. Exact solutions of nonlinear fractional order Sharma Tasso-Olever (STO) equation are expressed in terms of kink, travelling wave, periodic and solitary wave solutions. Modified Riemann-Liouville derivative and Fractional complex transform have been used for compatibility with fractional order sense. Solutions have been graphically simulated for understanding the physical aspects and importance of the method. A comparative discussion between our established results and the results obtained by existing ones is also presented. Our results clearly reveal that the proposed method is an effective, powerful and straightforward technique to work out new solutions of various types of differential equations of non-integer order in the fields of applied sciences and engineering.
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Geometry of classical particles on curved surfaces
Article
Full-text available
  • Feb 2017
In this paper we consider a particle moving on a curved surface. From a variational principle, we write the equation of motion and the constraining force, both in terms of the Darboux frame adapted to the trajectory, that involves geometric information of the surface. By deformation of the trajectory on the surface, the constraining force and equation of motion of the perturbation are obtained. We show that the transversal deformation follows a generalized Raychaudhuri equation that contains extrinsic information besides the geodesic curvature. Results in the case of surface with axial symmetry can be parametrized in terms of the angular momenta.
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Optical effects of some motion equations on quasi-frame with compatible Hasimoto map
Article
  • Sep 2021
  • OPTIK
In this study, we examine the relationship between some equations of motion and the equations obtained by considering the differential formula of the quasi-Hasimoto map and some physical issues. Here, we first obtain the necessary equations of the quasi-Hasimoto transformation as the quasi-frame equivalent of the Hasimoto map. Then, we examine the relations between the vortex filament equation and the Landau-Lifshitz equation, which are important equations with soliton solutions, with the quasi-Hasimoto map. Finally, we interpret the connection of the results obtained as a result of these examinations with some physical issues.
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Timelike spherical magnetic S N flux flows with Heisenberg spherical ferromagnetic spin with some solutions
Article
  • Mar 2021
  • OPTIK
In this paper, we first offer the approach of spherical magnetic Lorentz flux of spherical timelike SN−magnetic particle by the spherical frame in S12 spherical space. Eventually, we obtain some optical conditions of spherical t-magnetic Lorentz flux by using directional spherical fields. Moreover, we determine spherical magnetic Lorentz flux for spherical vector fields. Also, we give new construction for spherical curvatures of spherical timelike SN−magnetic particle flows by considering Heisenberg spherical ferromagnetic spin. Finally, magnetic flux surface is demonstrated in a static and uniform magnetic surface by using the analytical and numerical results.
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Approximate solutions for the inextensible Heisenberg antiferromagnetic flow and solitonic magnetic flux surfaces in the normal direction in Minkowski space
Article
  • Mar 2021
  • OPTIK
Motivated by recent researches in magnetic curves and their flows in different types of geometric manifolds and physical spacetime structures, we compute fractional Lorentz force equations associated with the magnetic n−lines in the normal direction in Minkowski space. Fractional evolution equations of magnetic n−lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their approximate solutions are investigated in terms of magnetic and geometric quantities via the conformable fractional derivative method. By considering arc-length and time-dependent orthogonal curvilinear coordinates, we finally determine the necessary and sufficient conditions that have to be satisfied by these quantities to define the Lorentz magnetic flux surfaces based on the inextensible Heisenberg antiferromagnetic flow model in Minkowski space.
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Binormal schrodinger system of wave propagation field of light radiate in the normal direction with q-HATM approach
Article
  • Feb 2021
  • OPTIK
In this manuscript, we study applications of flows of wave propagation in normal orientation of curved direction, which is concluded to be path of generated light radiate. Also, Coriolis phase is very productively used to illustrate the communication between geometric magnetic phase and parallel transport assumption of wave propagation field of evolving light radiate in normal orientation of curved direction. Thus, we present a consolidated geometric magnetic interpretation of binormal evolution of wave propagation field in the normal orientation of curved direction by the nonlinear fractional Schrodinger system with repulsive type. Additionally, we conclude obtain new numerical fractional solutions for nonlinear fractional Schrodinger system by using the q -HATM algorithm.
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Berry phase of the linearly polarized light wave along an optical fiber and its electromagnetic curves via quasi adapted frame
Article
  • Oct 2020
We review the geometric evolution of a linearly polarized light wave coupling into an optical fiber and the rotation of the polarization plane in a three dimensional ordinary space. We demonstrate that the evolution of a linearly polarized light wave is associated with the Berry phase or the geometric phase. We define other Fermi-Walker parallel transportation laws and connect them with the famous Rytov parallel transportation law for an electric field E , which is considered as the direction of the state of the linearly polarized light wave in the optical fiber in the 3D space. Later, we define a special class of magnetic curves called by quasi electromagnetic curves ( q E M -curves), which are generated by the electric field E along the linearly polarized monochromatic light wave propagating in the optical fiber. In this way, not only we define a special class of linearly polarized point-particles corresponding to q E M -curves of the electromagnetic field along with the optical fiber in the 3D space, but we also calculate, both numerically and analytically, the electromagnetic force, Poynting vector, energy-exchanges rate, optical angular and linear momentum, and optical magnetic-torque experienced by the linearly polarizable point-particles along with the optical fiber in the 3D space.
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