# Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Theorem**

**1.**

- (I)
- If${\omega}_{1}\left(t\right)={t}^{\gamma},$then${D}_{t}^{\alpha}{t}^{\gamma}=\frac{\Gamma \left(\gamma +1\right)}{\Gamma \left(\gamma +1-\alpha \right)}{t}^{\gamma -\alpha}$for$\gamma >0.$
- (II)
- ${D}_{t}^{\alpha}\left({\omega}_{1}\left(t\right){\omega}_{2}\left(t\right)\right)={\omega}_{2}\left(t\right){D}_{t}^{\alpha}{\omega}_{1}\left(t\right)+{\omega}_{1}\left(t\right){D}_{t}^{\alpha}{\omega}_{2}\left(t\right).$
- (III)
- ${D}_{t}^{\alpha}{\omega}_{1}\left({\omega}_{2}\left(t\right)\right)=\frac{d}{d{\omega}_{2}}{\omega}_{1}\left({\omega}_{2}\left(t\right)\right){D}_{t}^{\alpha}{\omega}_{2}\left(t\right)={D}_{{\omega}_{2}}^{\alpha}{\omega}_{1}\left({\omega}_{2}\left(t\right)\right){\left(\frac{d}{dt}{\omega}_{2}\left(t\right)\right)}^{\alpha}.$

## 3. The Extended Jacobi Elliptic Equation Method

#### 3.1. The Jacobi Elliptic Functions

$sn\left(u\right)\to \mathrm{tanh}u$, | $cn\left(u\right)\to \mathrm{sec}\mathrm{h}u$, | $dn\left(u\right)\to \mathrm{sec}\mathrm{h}u$, |

$ns\left(u\right)\to \mathrm{coth}u,$ | $nc\left(u\right)\to \mathrm{cosh}u,$ | $nd\left(u\right)\to \mathrm{cosh}u,$ |

$sc\left(u\right)\to \mathrm{sinh}u,$ | $sd\left(u\right)\to \mathrm{sinh}u,$ | $cd\left(u\right)\to 1$, |

$cs\left(u\right)\to \mathrm{csch}u,$ | $ds\left(u\right)\to \mathrm{csch}u$, | $dc\left(u\right)\to 1$. |

$sn\left(u\right)\to \mathrm{sin}u$, | $cn\left(u\right)\to \mathrm{cos}u$, | $dn\left(u\right)\to 1$, |

$ns\left(u\right)\to \mathrm{csc}u,$ | $nc\left(u\right)\to \mathrm{sec}u,$ | $nd\left(u\right)\to 1,$ |

$sc\left(u\right)\to \mathrm{tan}u,$ | $sd\left(u\right)\to \mathrm{sin}u,$ | $cd\left(u\right)\to \mathrm{cos}u$, |

$cs\left(u\right)\to \mathrm{cot}u,$ | $ds\left(u\right)\to \mathrm{csc}u$, | $dc\left(u\right)\to \mathrm{sec}u$. |

$c{n}^{2}\left(u\right)+s{n}^{2}\left(u\right)=1,$ | $d{n}^{2}\left(u\right)=1-{\delta}^{2}s{n}^{2}\left(u\right),$ |

$n{s}^{2}\left(u\right)-c{s}^{2}\left(u\right)=1,$ | $n{d}^{2}\left(u\right)=1+{\delta}^{2}s{d}^{2}\left(u\right),$ |

$n{c}^{2}\left(u\right)-s{c}^{2}\left(u\right)=1,$ | $c{d}^{2}\left(u\right)+\left(1-{\delta}^{2}\right)s{d}^{2}\left(u\right)=1,$ |

$n{s}^{2}\left(u\right)-d{s}^{2}\left(u\right)={\delta}^{2},$ | $d{c}^{2}\left(u\right)-\left(1-{\delta}^{2}\right)s{c}^{2}\left(u\right)=1,$ |

$d{s}^{2}\left(u\right)-c{s}^{2}\left(u\right)=1-{\delta}^{2},$ | $d{c}^{2}\left(u\right)-\left(1-{\delta}^{2}\right)n{c}^{2}\left(u\right)={\delta}^{2},$ |

${\delta}^{2}\left(c{n}^{2}\left(u\right)-1\right)-d{n}^{2}\left(u\right)=1,$ | ${\delta}^{2}c{d}^{2}\left(u\right)+\left(1-{\delta}^{2}\right)n{d}^{2}\left(u\right)=1.$ |

${\left(snu\right)}^{\prime}=cn\left(u\right)dn\left(u\right)$, | ${\left(cnu\right)}^{\prime}=-sn\left(u\right)dn\left(u\right)$, | ${\left(dnu\right)}^{\prime}=-{\delta}^{2}sn\left(u\right)cn\left(u\right)$, |

${\left(nsu\right)}^{\prime}=-cs\left(u\right)ds\left(u\right),$ | ${\left(ncu\right)}^{\prime}=sc\left(u\right)dc\left(u\right),$ | ${\left(ndu\right)}^{\prime}={\delta}^{2}cd\left(u\right)sd\left(u\right),$ |

${\left(scu\right)}^{\prime}=nc\left(u\right)dc\left(u\right),$ | ${\left(sdu\right)}^{\prime}=nd\left(u\right)cd\left(u\right),$ | ${\left(cdu\right)}^{\prime}=\left({\delta}^{2}-1\right)sd\left(u\right)nd\left(u\right)$, |

${\left(csu\right)}^{\prime}=-ns\left(u\right)ds\left(u\right),$ | ${\left(dsu\right)}^{\prime}=-ns\left(u\right)cs\left(u\right)$, | ${\left(dcu\right)}^{\prime}=\left(1-{\delta}^{2}\right)nc\left(u\right)sc\left(u\right)$. |

#### 3.2. Extended Jacobi Elliptic Function Expansion Method

## 4. Solving the Space–Time Fractional MTM

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Beiser, A. Concepts of Modern Physics, 6th ed.; McGraw-Hill: New York, NY, USA, 1994; ISBN 978-0072448481. [Google Scholar]
- Kiselev, V.; Shnir, Y.; Tregubovich, A. Introduction to Quantum Field Theory, 1st ed.; CRC Press: Boca Raton, FL, USA, 2000. [Google Scholar] [CrossRef]
- Thirring, W. A soluble relativistic field theory. Ann. Phys.
**1958**, 3, 91–112. [Google Scholar] [CrossRef] - Kondo, K.-I. Bosonization and Duality of Massive Thirring Model. Prog. Theor. Phys.
**1995**, 94, 899–914. [Google Scholar] [CrossRef] [Green Version] - Kondo, K.-I. Thirring model as a gauge theory. Nucl. Phys. B
**1995**, 450, 251–266. [Google Scholar] [CrossRef] [Green Version] - Joshi, N.; Pelinovsky, D.E. Integrable semi-discretization of the massive Thirring system in laboratory coordinates. J. Phys. A Math. Theor.
**2019**, 52, 03LT01. [Google Scholar] [CrossRef] [Green Version] - Laskin, N. Fractional Quantum Mechanics; World Scientific Publishing Co. Pte., Ltd.: Singapore, 2018. [Google Scholar]
- Lu, B. Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations. Phys. Lett. A
**2012**, 376, 2045–2048. [Google Scholar] [CrossRef] - Zahran, E.H.M.; Khater, M.M.A. Modified extended tanh-function method and its applications to the Bogoyavlenskii equation. Appl. Math. Model.
**2016**, 40, 1769–1775. [Google Scholar] [CrossRef] - Dubey, V.P.; Kumar, R.; Kumar, D.; Khan, I.; Singh, J. An efficient computational scheme for nonlinear time fractional systems of partial differential equations arising in physical sciences. Adv. Differ. Equat.
**2020**, 2020, 46. [Google Scholar] [CrossRef] - Gaber, A.A.; Aljohani, A.F.; Ebaid, A.; Machado, J.T. The generalized Kudryashov method for nonlinear space–time fractional partial differential equations of Burgers type. Nonlinear Dyn.
**2019**, 95, 361–368. [Google Scholar] [CrossRef] - Benfatto, G.; Falco, P.; Mastropietro, V. Functional Integral Construction of the Massive Thirring model: Verification of Axioms and Massless Limit. Commun. Math. Phys.
**2007**, 273, 67–118. [Google Scholar] [CrossRef] [Green Version] - Bergknoff, H.; Thacker, H.B. Structure and solution of the massive Thirring model. Phys. Rev. D
**1979**, 19, 3666–3681. [Google Scholar] [CrossRef] - Fujita, T.; Sekiguchi, Y.; Yamamoto, K. A New Interpretation of Bethe Ansatz Solutions for Massive Thirring Model. Ann. Phys.
**1997**, 255, 204–227. [Google Scholar] [CrossRef] [Green Version] - Delépine, D.; Felipe, R.G.; Weyers, J. Equivalence of the sine-Gordon and massive Thirring models at finite temperature. Phys. Lett. B
**1998**, 419, 296–302. [Google Scholar] [CrossRef] [Green Version] - Aydogmus, F.; Tosyali, E. Numerical Analysis of Thirring Model under White Noise. J. Phys. Conf. Ser.
**2015**, 633, 012022. [Google Scholar] [CrossRef] - Bañuls, M.C.; Cichy, K.; Kao, Y.-J.; Lin, C.-J.D.; Lin, Y.-P.; Tan, D.T.-L. Phase structure of the (1+1)-dimensional massive Thirring model from matrix product states. Phys. Rev. D
**2019**, 100, 094504. [Google Scholar] [CrossRef] [Green Version] - Guo, L.; Wang, L.; Cheng, Y.; He, J. High-order rogue wave solutions of the classical massive Thirring model equations. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 52, 11–23. [Google Scholar] [CrossRef] - Arafa, A.A.M.; Hagag, A.M.S. Q-homotopy analysis transform method applied to fractional Kundu–Eckhaus equation and fractional massive Thirring model arising in quantum field theory. Asian Eur. J. Math.
**2019**, 12, 1950045. [Google Scholar] [CrossRef] - Al-Smadi, M.; Abu Arqub, O.; Hadid, S. Approximate solutions of nonlinear fractional Kundu-Eckhaus and coupled fractional massive Thirring equations emerging in quantum field theory using conformable residual power series method. Phys. Scr.
**2020**, 95, 105205. [Google Scholar] [CrossRef] - Al-Smadi, M.; Freihat, A.; Khalil, H.; Momani, S.; Khan, R.A. Numerical multistep approach for solving fractional partial dif-ferential equations. Int. J. Comput. Meth.
**2017**, 14, 1750029. [Google Scholar] [CrossRef] - Hasan, S.; El-Ajou, A.; Hadid, S.; Al-Smadi, M.; Momani, S. Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system. Chaos Solitons Fractals
**2020**, 133, 109624. [Google Scholar] [CrossRef] - Al-Smadi, M.; Abu Arqub, O. Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates. Appl. Math. Comput.
**2019**, 342, 280–294. [Google Scholar] [CrossRef] - Al-Smadi, M.; Abu Arqub, O.; Momani, S. Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense. Phys. Scr.
**2020**, 95, 075218. [Google Scholar] [CrossRef] - Al-Smadi, M.; Abu Arqub, O.; Hadid, S. An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative. Commun. Theor. Phys.
**2020**, 72, 085001. [Google Scholar] [CrossRef] - Al-Smadi, M.; Abu Arqub, O.; Gaith, M. Numerical simulation of telegraph and Cattaneo fractional-type models using adaptive reproducing kernel framework. Math. Methods Appl. Sci.
**2020**, 44, 8472–8489. [Google Scholar] [CrossRef] - Nairat, M.; Shqair, M.; Alhalholy, T. Cylindrically Symmetric Fractional Helmholtz Equation. Appl. Math.
**2019**, 19, 708–717. [Google Scholar] - Shqair, M. Developing a new approaching technique of homotopy perturbation method to solve two-group reflected cy-lindrical reactor. Results Phys.
**2019**, 12, 1880–1887. [Google Scholar] [CrossRef] - Wazwaz, A.-M. A variety of multiple-soliton solutions for the integrable (4+1)-dimensional Fokas equation. Waves Random Complex Media
**2021**, 31, 46–56. [Google Scholar] [CrossRef] - Wen, X.; Feng, R.; Lin, J.; Liu, W.; Chen, F.; Yang, Q. Distorted light bullet in a tapered graded-index waveguide with PT symmetric potentials. Optik
**2021**, 248, 168092. [Google Scholar] [CrossRef] - Fang, J.J.; Mou, D.S.; Zhang, H.C.; Wang, Y.Y. Discrete fractional soliton dynamics of the fractional Ablowitz-Ladik model. Optik
**2021**, 228, 166186. [Google Scholar] [CrossRef] - Liu, C.-S. Counterexamples on Jumarie’s two basic fractional calculus formulae. Commun. Nonlinear Sci. Numer. Simul.
**2015**, 22, 92–94. [Google Scholar] [CrossRef] - Liu, C.-S. Counterexamples on Jumarie’s three basic fractional calculus formulae for non-differentiable continuous functions. Chaos Solitons Fractals
**2018**, 109, 219–222. [Google Scholar] [CrossRef] [Green Version] - Tarasov, V.E. No nonlocality. No fractional derivative. Commun. Nonlinear Sci. Numer. Simulat.
**2018**, 62, 157–163. [Google Scholar] [CrossRef] [Green Version] - Tarasov, V.E. No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear. Sci. Numer. Simulat.
**2013**, 18, 2945–2948. [Google Scholar] [CrossRef] [Green Version] - Yu, L.-J.; Wu, G.-Z.; Wang, Y.-Y.; Chen, Y.-X. Traveling wave solutions constructed by Mittag–Leffler function of a (2+1)-dimensional space-time fractional NLS equation. Results Phys.
**2020**, 17, 103156. [Google Scholar] [CrossRef] - Wu, G.-Z.; Yu, L.-J.; Wang, Y.-Y. Fractional optical solitons of the space-time fractional nonlinear Schrödinger equation. Int. J. Light Electron Opt.
**2020**, 207, 164405. [Google Scholar] [CrossRef] - Das, A.; Ghosh, N. Bifurcation of traveling waves and exact solutions of Kadomtsev–Petviashvili modified equal width equation with fractional temporal evolution. Comput. Appl. Math.
**2019**, 38, 9. [Google Scholar] [CrossRef] - Jiang, J.; Feng, Y.; Li, S. Improved Fractional Subequation Method and Exact Solutions to Fractional Partial Differential Equations. J. Funct. Spaces
**2020**, 2020, 5840920. [Google Scholar] [CrossRef] - Li, C.; Guo, Q.; Zhao, M. On the solutions of (2+1)-dimensional time-fractional Schrödinger equation. Appl. Math. Lett.
**2019**, 94, 238–243. [Google Scholar] [CrossRef] - Aksoy, E.; Bekir, A.; Çevikel, A.C. Study on Fractional Differential Equations with Modified Riemann–Liouville Derivative via Kudryashov Method. Int. J. Nonlinear Sci. Numer. Simul.
**2019**, 20, 511–516. [Google Scholar] [CrossRef]

**Figure 1.**The 3D plots of ${\left|{F}_{1}\left(x,t\right)\right|}^{2}$ and ${\left|{G}_{1}\left(x,t\right)\right|}^{2}$ with the parameters ${r}_{1}={r}_{2}=1,{k}_{1}={k}_{2}=1,{q}_{0}=0.6,\text{}{p}_{2}=-2,\text{}{q}_{3}=0.8,\rho =1$ and $\tau =0$ for various $\alpha $ values: (

**a**)${\left|{F}_{1}\right|}^{2},\alpha =1$, (

**b**)${\left|{G}_{1}\right|}^{2},\alpha =1,$ (

**c**) ${\left|{F}_{1}\right|}^{2},\alpha =0.75$ and (

**d**) ${\left|{G}_{1}\right|}^{2},\alpha =0.75.$

**Figure 2.**The 3D plots of the real and imaginary parts of ${F}_{2}\left(x,t\right)$ and with the parameters ${r}_{1}={r}_{2}=1,\text{}{k}_{1}={k}_{2}=1,{q}_{0}=0.2,\text{}{p}_{2}=-1,\text{}{q}_{3}=0.5$ and $\rho =\tau =1$ for the fractional order $\alpha =0.85$: (

**a**) $Re\left[{F}_{2}\left(x,t\right)\right]$, (

**b**) $Im\left[{F}_{2}\left(x,t\right)\right],$ (

**c**) $Re\left[{G}_{2}\left(x,t\right)\right]$ and (

**d**) $Im\left[{G}_{2}\left(x,t\right)\right].$

**Figure 3.**The 3D plots of ${\left|{F}_{5}\left(x,t\right)\right|}^{2}$ and ${\left|{G}_{5}\left(x,t\right)\right|}^{2}$ with the parameters ${r}_{1}={r}_{2}=-1,{k}_{1}={k}_{2}=1,{q}_{0}=0.05,\text{}{p}_{3}=-0.3,\text{}{q}_{1}=2,\rho =1$ and $\tau =0$ for various $\alpha $ values: (

**a**) ${\left|{F}_{5}\right|}^{2},\alpha =1$, (

**b**) ${\left|{G}_{5}\right|}^{2},\alpha =1,$ (

**c**) ${\left|{F}_{5}\right|}^{2},\alpha =0.75$ and (

**d**) ${\left|{G}_{5}\right|}^{2},\alpha =0.75.$

**Figure 4.**The 3D plots of the real and imaginary parts of ${F}_{6}\left(x,t\right)$ and ${G}_{6}\left(x,t\right)$ with the parameters ${r}_{1}={r}_{2}=-1,{k}_{1}={k}_{2}=1,{q}_{0}=-2.5,\text{}{p}_{3}=0,\text{}{q}_{1}=-1,\rho =1$ and $\tau =0$ for the fractional order $\alpha =1$: (

**a**) $Re\left[{F}_{6}\left(x,t\right)\right]$, (

**b**) $Re\left[{G}_{6}\left(x,t\right)\right],$ (

**c**) $Im\left[{F}_{6}\left(x,t\right)\right]$ and (

**d**) $Im\left[{G}_{6}\left(x,t\right)\right].$

**Figure 5.**The 3D plots of the real and imaginary parts of ${F}_{7}\left(x,t\right)$ and ${G}_{7}\left(x,t\right)$ with the parameters ${r}_{1}={r}_{2}=0.3,{k}_{1}={k}_{2}=1,{q}_{0}=0.9,\text{}{p}_{3}=4,\text{}{q}_{1}=5/7,\rho =1$ and $\tau =0$ for the fractional order $\alpha =0.65$: (

**a**) $Re\left[{F}_{7}\left(x,t\right)\right]$, (

**b**) $Re\left[{G}_{7}\left(x,t\right)\right],$ (

**c**) $Im\left[{F}_{7}\left(x,t\right)\right]$ and (

**d**) $Im\left[{G}_{7}\left(x,t\right)\right].$

**Figure 6.**The 3D plots of the real and imaginary parts of ${F}_{10}\left(x,t\right)$ and ${G}_{10}\left(x,t\right)$ with the parameters ${r}_{1}={r}_{2}=0.3,{k}_{1}={k}_{2}=1,{q}_{0}=0.9,\text{}{p}_{2}=1,\text{}{q}_{1}=-0.2,\rho =0.5$ and $\tau =0$ for the fractional order $\alpha =0.9$: (

**a**) $Re\left[{F}_{10}\left(x,t\right)\right]$, (

**b**) $Re\left[{G}_{10}\left(x,t\right)\right],$ (

**c**) $Im\left[{F}_{10}\left(x,t\right)\right]$ and (

**d**) $Im\left[{G}_{10}\left(x,t\right)\right].$

**Figure 7.**The 3D plots of ${\left|{F}_{12}\left(x,t\right)\right|}^{2}$ and ${\left|{G}_{12}\left(x,t\right)\right|}^{2}$ with the parameters ${r}_{1}={r}_{2}=-3,{k}_{1}={k}_{2}=1,{q}_{0}=7,\text{}{p}_{2}=2,\text{}{q}_{1}=-5,\rho =0.5$ and $\tau =0$ for various $\alpha $ values: (

**a**) ${\left|{F}_{12}\right|}^{2},\alpha =0.95$, (

**b**) ${\left|{G}_{12}\right|}^{2},\alpha =0.95,$ (

**c**) ${\left|{F}_{12}\right|}^{2},\alpha =0.75$ and (

**d**) ${\left|{G}_{12}\right|}^{2},\alpha =0.75.$

**Figure 8.**The 3D plots of the real and imaginary parts of ${F}_{13}\left(x,t\right)$ and ${G}_{13}\left(x,t\right)$ with the parameters ${r}_{1}={r}_{2}=-3,{k}_{1}={k}_{2}=1,{q}_{0}=7,\text{}{p}_{2}=2,\text{}{q}_{1}=-5,\rho =0.5,$ and $\tau =0.1$ for the fractional order $\alpha =0.8$: (

**a**) $Re\left[{F}_{13}\left(x,t\right)\right]$, (

**b**) $Re\left[{G}_{13}\left(x,t\right)\right],$ (

**c**) $Im\left[{F}_{13}\left(x,t\right)\right],$ (

**d**) $Im\left[{G}_{13}\left(x,t\right)\right].$

**Figure 9.**The 3D plots of the real and imaginary parts of ${F}_{14}\left(x,t\right)$ and ${G}_{14}\left(x,t\right)$ with the parameters ${r}_{1}={r}_{2}=0.5,{k}_{1}={k}_{2}=1,{q}_{0}=10,\text{}{p}_{2}=0.2,\text{}{q}_{1}=1,\rho =1$ and $\tau =0.1$ for the fractional order $\alpha =0.75$: (

**a**) $Re\left[{F}_{14}\left(x,t\right)\right]$, (

**b**) $Re\left[{G}_{14}\left(x,t\right)\right],$ (

**c**) $Im\left[{F}_{14}\left(x,t\right)\right]$ and (

**d**) $Im\left[{G}_{14}\left(x,t\right)\right].$

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shqair, M.; Alabedalhadi, M.; Al-Omari, S.; Al-Smadi, M.
Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method. *Fractal Fract.* **2022**, *6*, 252.
https://doi.org/10.3390/fractalfract6050252

**AMA Style**

Shqair M, Alabedalhadi M, Al-Omari S, Al-Smadi M.
Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method. *Fractal and Fractional*. 2022; 6(5):252.
https://doi.org/10.3390/fractalfract6050252

**Chicago/Turabian Style**

Shqair, Mohammed, Mohammed Alabedalhadi, Shrideh Al-Omari, and Mohammed Al-Smadi.
2022. "Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method" *Fractal and Fractional* 6, no. 5: 252.
https://doi.org/10.3390/fractalfract6050252