Research Article

In this academic investigation, an innovative mapping approach is applied to complex three coupled Maccari’s system to unveil novel soliton solutions. This is achieved through the utilization of M-Truncated fractional derivative with employing the new mapping method and computer algebraic syatem (CAS) such as Maple. The derived solutions in the form of hyperbolic and trigonometric functions. Our study elucidates a variety of soliton solutions such as periodic, singular, dark, kink, bright, dark-bright solitons solutions. To facilitate comprehension, with certain solutions being visually depicted through 2-dimensional, contour, 3-dimensional, and phase plots depicting bifurcation characteristics utilizing Maple software. Furthermore, the incorporation of M-Truncated derivative enables a more extensive exploration of solution patterns. Our study establishes a connection between computer science and soliton physics, emphasizing the pivotal role of soliton phenomena in advancing simulations and computational modeling. Analytical solutions are subsequently generated through the application of the new mapping method. Following this, a thorough examination of the dynamic nature of the equation is conducted from various perspectives. In essence, understanding the dynamic characteristics of systems is of great importance for predicting outcomes and advancing new technologies. This research significantly contributes to the convergence of theoretical mathematics and applied computer science, emphasizing the crucial role of solitons in scientific disciplines.

Complex three coupled Maccari’s system A new mapping method; Soliton patterns Bifurcation; M-Truncated fractional derivative

VSB-Technical University of Ostrava

e-INFRA CZ (ID:90254).

The Authors are thankful to the Czech Ministry of Education, Youth, and Sports for their assistance through the e-INFRA CZ (ID:90254).

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- X. Zeng, X. Y., 2008 A new mapping method and its applications to nonlinear partial differential equations. Phys. Lett. A 372: 6602–6607.
- Zaslavsky, G., 2002 Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371: 461–580.
- Zayed, E. M. and K. A. AlurrÖ, 2017 Solitons and other solutions for two nonlinear Schrödinger equations using the new mapping method. Optik 11: 132–148.
- Zayed, E. M. E. and K. A. E. AlurrÖ, 2015 A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines, Chaos, Solitons and Fractals. . Mathematics 78: 148–155.
- Zheng, B. and Q. Feng, 2014 The Jacobi elliptic equation method for solving fractional partial di§erential equations . Abs. Appl. Anal. 2014.
- Zhu, L. Z. X.-Y.,W. and M. Gao, 2023 Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity. Fractal Fract. 7.

Year 2024,
Volume: 6 Issue: 3, 180 - 191, 31.07.2024
### Abstract

### Project Number

### References

e-INFRA CZ (ID:90254).

- A. Filiz, M. E. and A. Sonmezoglu, 2014 F-expansion method and new exact- solution of the Schrödinger-KdV equation . Sci. World J. 2014.
- Alabedalhadi, A.-O. S.-A.-S.-M., M. and S. Alhazmi, 2023 Traveling Wave Solutions for Time-Fractional mKdV-ZK Equation of Weakly Nonlinear Ion-Acoustic Waves in Magnetized Electron– Positron Plasma . Symmetry 15.
- Das, N. and S. Saha Ray, 2022 Dispersive optical soliton wave solutions for the time-fractional perturbed nonlinear Schrödinger equation with truncated M-fractional conformable derivative in the nonlinear optical fibers. Opt. Quantum Electron 54.
- Das, N. and S. Saha Ray, 2023 Dispersive optical soliton solutions of the (2+1)-dimensional cascaded system governing by coupled nonlinear Schrödinger equation with Kerr law nonlinearity in plasma. Opt. Quantum Electron 55.
- E. M. Zayed, R. M. S. A. B. Y. Y. A. S. A., M. E. Alngar and H. M. Alshehr, 2022 Optical solitons having Kudryashovís self-phase modulation with multiplicative white noise via ItÙ Calculus using new mapping approach. Optik 264.
- Ellahi, M.-D. S., R. and U. Khan, 2018 Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method. Results in Phys 8: 114–120.
- Emad, Z.-M. S. S. M.-A. M., H.M. and A. Lanre, 2021 Exact propagation of the isolated waves model described by the three coupled nonlinear Maccari’s system with complex structure. . Int. J. Mod. Phys. B 35.
- Ge, Z.-M. and C.-Y. Ou, 2008 Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal. Chaos Solitons Fractals 35: 705–717.
- Khater, B. N. K., M.M.A. Ghanbari and D. Kumar, 2020 Novel exact solutions of the fractional Bogoyavlensky–Konopelchenko equation involving the Atangana-Baleanu-Riemann derivative. Alex. Eng. J. 59: 2957–2967.
- Mohammed, E.-M. M. M. A.-A. E. B. M., W.W. and A. Abouelregal, 2023 Effects of M-Truncated Derivative and Multiplicative Noise on the Exact Solutions of the Breaking Soliton Equation. . Symmetry 15.
- Naeem, R.-H. K. A. S.-R., M. and S. Zaland, 2022 Analysis of the fuzzy fractional-order solitary wave solutions for the KdV equation in the sense of Caputo-Fabrizio derivative. . J. Math. 2022: 2957–2967.
- Rafiq, M.-A. I. M., M.N. and M. Kamran, 2022 New traveling wave solutions for space-time fractional modified equal width equation with beta derivative. Phys. Lett. A 446: 411–425.
- Saha Ray, S. and N. Das, 2022 Novel optical soliton solutions for time-fractional resonant nonlinear Schrödinger equation in optical fiber. . Mod.Phys. Lett. B 36.
- Senol, M., 2020 New analytical solutions of fractional symmetric regularized-long-wave equation. Rev. Mex. Fís. 66: 297–307.
- Vanterler, D. C. E. O.-D., J.; Sousa, 2018 A new truncated Mfractional derivative type unifying some fractional derivative types with classical properties. Int. J. Anal. Appl 16: 83–96.
- W. B. Rabie, H. M. A. and W. Hamdy, 2023 Exploration of new optical solitons in magnetooptical waveguide with coupled system of nonlinear BiswasñMilovic equation via Kudryashovís law using extended F-expansion method. . Mathematics 11.
- Wang, S. C.-Q. L.-X.-Q., Y.-Y. and J.-G. Li, 2018 Nonautonomous solitons for an extended forced Korteweg-de Vries equation with variable coefficients in the fluid or plasma. Waves Random Complex Media 3: 411–425.
- X. Zeng, X. Y., 2008 A new mapping method and its applications to nonlinear partial differential equations. Phys. Lett. A 372: 6602–6607.
- Zaslavsky, G., 2002 Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371: 461–580.
- Zayed, E. M. and K. A. AlurrÖ, 2017 Solitons and other solutions for two nonlinear Schrödinger equations using the new mapping method. Optik 11: 132–148.
- Zayed, E. M. E. and K. A. E. AlurrÖ, 2015 A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines, Chaos, Solitons and Fractals. . Mathematics 78: 148–155.
- Zheng, B. and Q. Feng, 2014 The Jacobi elliptic equation method for solving fractional partial di§erential equations . Abs. Appl. Anal. 2014.
- Zhu, L. Z. X.-Y.,W. and M. Gao, 2023 Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity. Fractal Fract. 7.

There are 23 citations in total.

Primary Language | English |
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Subjects | Complex Systems in Mathematics, Theoretical and Applied Mechanics in Mathematics, Dynamical Systems in Applications, Applied Mathematics (Other) |

Journal Section | Research Articles |

Authors | |

Project Number | e-INFRA CZ (ID:90254). |

Publication Date | July 31, 2024 |

Submission Date | January 5, 2024 |

Acceptance Date | April 4, 2024 |

Published in Issue | Year 2024 Volume: 6 Issue: 3 |

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science

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