In this article, the $(G^{{}^{\prime }}/G)$ -expansion method is used for the analytical solutions of fractional-order Klein-Gordon and Gas Dynamics equations. The fractional derivatives are defined in the term of Jumarie’s operator. The proposed method is based on certain variable transformation, which transforms the given problems into ordinary differential equations. The solution of resultant ordinary differential equation can be expressed by a polynomial in $(G^{{}^{\prime }}/G)$ , where $G=G(\xi )$ satisfies a second order linear ordinary differential equation. In this paper, $(G^{{}^{\prime }}/G)$ -expansion method will represent, the travelling wave solutions of fractional-order Klein-Gordon and Gas Dynamics equations in the term of trigonometric, hyperbolic and rational functions.

The current research focusses on the establishment of an analytical approach known as the Riccati Modified Extended Simple equation Method (RMESEM) for the development and assessment of optical soliton solutions in two important Kairat equations. These models are known as Kairat-X equation (K-XE) and the Kairat-II equation (K-IIE), which describe the trajectory of optical pulses in optical fibres. Using RMESEM, the soliton solutions in five families–the periodic, rational, hyperbolic, rational-hyperbolic, and exponential functional families–are achieved for the targeted models. A set of 3D, 2D, and contour visualisations are presented to visually illustrate the dynamics of some produced optical soliton solutions which demonstrates that the due to the axial-periodic perturbation, the optical soliton solutions exhibit fractal phenomena in the realm of K-IIE whereas in the setting of K-XE the optical solitons adopt the form of kink solitons such as solitary kink, lump-type kink, dromion and periodic kink soliton structures. Moreover, our suggested RMESEM illustrates its usefulness by building a multitude of optical soliton solutions, providing valuable insights into the dynamics of the targeted models and indicating potential uses in addressing other nonlinear models.

The current study introduces the generalised New Extended Direct Algebraic Method (gNEDAM) for producing and examining propagation of kink soliton solutions within the framework of the Conformable Kolmogorov–Petrovskii–Piskunov Equation (CKPPE), which entails conformable fractional derivatives into account. The primary justification around employing conformable derivatives in this study is their special ability to comply with the chain rule, allowing for in the solution of aimed nonlinear model. The CKPPE is a crucial model for a number of disciplines, such as mathematical biology, reaction-diffusion mechanisms, and population increase. CKPPE is transformed into a Nonlinear Ordinary Differential Equation by the proposed gNEDAM, and many kink soliton solutions are found by applying the series form solution. These kink soliton solutions shed light on propagation mechanisms within the framework of the CKPPE model. Furthermore, our research offers multiple graphical depictions that facilitate the examination and analysis of the propagation patterns of the identified kink soliton solutions. Through the integration of mathematical biology and reaction-diffusion principles, our research broadens our comprehension of intricate occurrences in various academic domains.

In this scholarly article, we investigate the complex structured (3+1)-dimensional Fractional Heisenberg Ferromagnetic Spin Chain equation (FHFSCE) with conformable fractional derivatives. We develop a diverse glut of soliton solutions using an improved version of the G ′ G -expansion method, namely the r + G ′ G -expansion method. The constraints for the existence of these solutions are painstakingly explained. Our findings are clearly communicated through a number of 3D and 2D graphical representations displaying periodic, multiple periodic, kink, and shock soliton solutions. These soliton solutions are expressed in several mathematical function forms, such as hyperbolic, trigonometric, and rational functions. Our findings support the suggested method’s efficacy as a powerful symbolic algorithm for discovering innovative soliton solutions within nonlinear evolution systems.

In this research, we aim to construct and examine optical soliton solutions for the complex structured Conformable Perturbed Radhakrishnan-Kundu-Lakshmanan Model (CPRKLM) using the Generalized-Kudryashov-Auxiliry Jacobian Method (GKAJM). The current study is notable for its thorough examination and for shedding insight on the chaotic behavior of families of localized optical soliton. Through the creation of 3D and contour visualizations that effectively capture the chaotic behaviors shown by these solitons, we are able to demonstrate that the optical solitons exhibit two distinct forms of perturbations: axial and periodic. Our research stimulates improvements in data processing tools and optical equipment, with consequences for communication networks and nonlinear fiber optics. Through a deeper understanding of optical solitons and their applications, this work also makes a substantial contribution to the discipline of nonlinear optics.

In this study, we consider new optical soliton solutions of one of the most important non-integrable model arising in optical fibres, namely Nonlinear Helmholtz equations (NHEs) that describes transverse interactions, transmission of coupled waves and optical solitons’ propagation in the field of fiber optics. We apply an adapted method to obtain some novel plethora of optical quasi-periodic soliton solutions. These solutions are presented in the shape of exponential, hyperbolic, trigonometric and rational functions. A set of 3D visualization, contour plots and 2D curves of these solutions physical relevance are presented with implications for the nonlinear optics. These figures also reveal that the established optical solitons exhibit quasi-periodicity due to the combination of linear periodic and axial perturbations, and that the presence of quasi-periodical perturbations of the solitons leads to the formation of the fractal-like structures. We also study the chaotic/periodic and bifurcation behavior, associated with the model, in the light of Hamiltonian analysis, as a consequence, we find positive results of the quasi-periodicity and fractal-like structures in the systems under consideration. Apart from offering novel analytical perspectives for dealing with the coupled NHEs, the present results would also be a concrete contribution to the understanding the soliton wave dynamics in complicated nonlinear media.