Bifurcation, quasi-periodic, and chaotic analysis of the conformable fractional paraxial wave equation using the rational expansion method
Source abstract
Abstract
In this study, the conformable space–time fractional paraxial wave equation, which describes the optical pulse propagation through nonlinear media, is investigated to gain insight into intricate nonlinear phenomena. The objective of the study is to identify novel exact soliton solutions of the concerned equation with the implication of a robust analytical approach—the rational (G′/G)-expansion scheme. The suggested method offers a variety of hyperbolic, trigonometric, and rational exact solutions to the governing equation. Assorted types of solitons, including Kuznetsov-Ma breather type, periodic, flat-soliton, bell-shaped, kink, shock wave, and singular breather, are obtained in this study by selecting appropriate values for the free parameters included in analytical solutions. We utilize the Mathematica software to plot three- and two-dimensional graphs and contours of the obtained solitons. The bifurcation theory is used to analyze the dynamical behavior of the planar dynamic system produced from the considered equation via the Galilean transformation. An external periodic force is introduced to construct a perturbed nonlinear dynamical system that allows us to observe the quasi-periodic and chaotic nature of the system. The phase portraits and time series are presented to support the consistency of the results.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Nonlinear Waves and Solitons · Fractional Differential Equations Solutions · Nonlinear Photonic Systems
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
Affiliations
Department of Mathematics, Uttara University 1 , Uttara,
Department of Mathematics, College of Science, Taif University 2 , P.O. Box 11099, 21944 Taif,
Department of Mathematics, Khulna University of Engineering and Technology 3 , Khulna,
Department of Applied Mathematics, University of Rajshahi 4 , Rajshahi,
Funding
Taif University
References (49)
Show source-provided references
1. A new approach to generalized fractional derivatives
2. Linear models of dissipation whose Q is almost frequency independent--II10.1111/j.1365-246x.1967.tb02303.x ↗
3. On the Riemann–Liouville fractional calculus and some recent applications10.1142/s0218348x95000497 ↗
4. A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative10.1016/j.chaos.2020.109705 ↗
5. Fractional differential equations with Atangana–Baleanu fractional derivative: Analysis and applications10.1016/j.csfx.2019.100013 ↗
6. Controllability, observability and fractional linear-quadratic problem for fractional linear systems with conformable fractional derivatives and some applications10.1007/s40435-022-00977-7 ↗
7. New perspective to the fractal Konopelchenko–Dubrovsky equations with M-truncated fractional derivative10.1142/s021988782350072x ↗
8. Interaction of solitons in a collisionless plasma and the recurrence of initial states10.1103/physrevlett.15.240 ↗
9. Soliton: A dispersion-less solution with existence and its types10.1016/j.heliyon.2022.e12122 ↗
10. New closed form solutions of some nonlinear pseudo-parabolic models via a new extended direct algebraic method10.2478/ijmce-2024-0004 ↗
11. Applying the new extended direct algebraic method to solve the equation of obliquely interacting waves in shallow waters10.1007/s11802-020-4135-8 ↗
12. A study on the compatibility of the generalized Kudryashov method to determine wave solutions10.1016/j.jppr.2020.12.001 ↗
13. The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative10.31349/revmexfis.66.771 ↗
14. The (G′/G)-expansion method for the nonlinear lattice equations10.1016/j.cnsns.2012.01.009 ↗
15. The exact solutions of stochastic fractional-space Kuramoto–Sivashinsky equation by using (G′/G)-expansion method10.3390/math9212712 ↗
16. Traveling wave solutions of Benny–Luke equation via the enhanced (G′/G)-expansion method10.1016/j.asej.2017.03.018 ↗
17. Immeasurable soliton solutions and enhanced (G′/G)-expansion method10.1016/j.physo.2021.100086 ↗
18. Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized (G′/G)-expansion method10.1515/ijnsns-2020-0153 ↗
19. Using the improved exp−ϕξ-expansion method to find the soliton solutions of the nonlinear evolution equation10.1140/epjp/s13360-021-01321-2 ↗
20. The improved exp−ϕξ-expansion method and new exact solutions of nonlinear evolution equations in mathematical physics10.1155/2019/4354310 ↗
21. Multiple soliton solutions for nonlinear differential equations with a new version of extended F-expansion method10.1007/s40010-020-00687-9 ↗
22. Generating optical solitons in the extended (3 + 1)-dimensional nonlinear Kudryashov’s equation using the extended F-expansion method10.1007/s11082-024-06787-9 ↗
23. Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method10.1016/j.rinp.2021.104228 ↗
24. Sine-Gordon expansion method to construct the solitary wave solutions of a family of 3D fractional WBBM equations10.1016/j.rinp.2022.105845 ↗
25. Wide-spectrum optical soliton solutions to the time-fractional cubic-quintic resonant nonlinear Schrödinger equation with parabolic law10.1016/j.rinp.2023.106862 ↗
26. Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G, 1/G)-expansion and (1/G′)-expansion techniques10.1016/j.rinp.2020.103409 ↗
27. Analytical soliton solutions of time-fractional higher-order Sasa–Satsuma equations: Nonlinear optics and beyond and the impact of fractional-order derivative10.1007/s11082-023-05823-4 ↗
28. On the exploration of solitary wave structures to the nonlinear Landau–Ginsberg–Higgs equation under improved F-expansion method10.1007/s11082-024-06458-9 ↗
29. Mathematical analysis of the dynamics of solitary wave solutions to the time-fractional thin-film ferroelectric materials model10.1016/j.rinp.2024.107621 ↗
30. Optical soliton solutions to the Fokas–Lenells model applying the ϕ6-model expansion approach10.1007/s11082-023-04771-3 ↗
31. The optical exact soliton solutions of Shynaray-IIA equation with ϕ6-model expansion approach10.1007/s11082-023-05814-5 ↗
32. Optical solitons of the extended Gerdjikov–Ivanov equation in DWDM system by extended simplest equation method10.18576/amis/140517 ↗
33. Novel periodic and optical soliton solutions for Davey–Stewartson system by generalized Jacobi elliptic expansion method10.1515/ijnsns-2021-0349 ↗
34. Illustration of the homotopy perturbation method to the modified nonlinear single degree of freedom system10.1016/j.chaos.2023.113481 ↗
35. Jacobi elliptic function expansion method for the improved modified Korteweg-de-Vries equation10.1007/s11082-022-04109-5 ↗
36. Analytical solutions of two space-time fractional nonlinear models using Jacobi elliptic function expansion method10.37256/cm.232021682 ↗
37. Dynamic behavior and optical soliton for the M-truncated fractional paraxial wave equation arising in a liquid crystal model10.3390/fractalfract8060348 ↗
38. New traveling wave solutions for paraxial wave equation via two integrating techniques10.1007/s11082-024-06589-z ↗
39. Dynamical exploration of optical soliton solutions for M-fractional Paraxial wave equation10.1371/journal.pone.0299573 ↗
40. Searching the new exact wave solutions to the beta-fractional paraxial nonlinear Schrödinger model via three different approaches10.1142/s0217979224501327 ↗
41. Modulation instability and comparative observation of the effect of fractional parameters on new optical soliton solutions of the paraxial wave model10.1007/s11082-024-06921-7 ↗
42. Dispersive optical soliton solutions to the truncated time M-fractional paraxial wave equation with its stability analysis10.1007/s11082-024-06663-6 ↗
43. Exact solution of paraxial wave dynamical model with Kerr Media by using ϕ6-model expansion technique10.1016/j.rinp.2022.105975 ↗
44. Optical solitons for paraxial wave equation in Kerr media10.1142/s0217984919500209 ↗
45. Elliptic function solutions, modulation instability and optical solitons analysis of the paraxial wave dynamical model with Kerr media10.1007/s11082-020-02637-6 ↗
46. Lump and optical dromions for paraxial nonlinear Schrödinger equation10.1142/s0217979221500788 ↗
47. Application of the rational (G′/G)-expansion method for solving some coupled and combined wave equations10.31801/cfsuasmas.884025 ↗
48. A rational (G′/G)-expansion method and its application to modified KdV-Burgers equation and the (2 + 1)-dimensional Boussinesq equation
49. Dynamics of exact solutions of nonlinear resonant Schrödinger equation utilizing conformable derivatives and stability analysis10.1140/epjd/s10053-023-00703-8 ↗