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This research paper's primary goal is to find fresh approaches to the Hirota–Maccari system. This system explains the dynamical features of the femto-second soliton pulse in single-mode fibers. The bright soliton, dark soliton, dark-bright soliton, dark singular, bright singular, periodic soliton, and singular solutions are developed utilizing the modified auxiliary equation technique. To make the physical significance of each unique solution clearer, it is mapped in both 2D and 3D. The primary Hirota–Maccari system is being verified by all new solutions, and the constraint condition is also provided. The obtained optical solitons may be important for the analysis of nonlinear processes in optic fiber communication and signal processing. [ABSTRACT FROM AUTHOR]

The study of the complex model associated with chaotic models is always the most complicated and fundamental in the current scientific environment. The primary goal of the current paper is to provide an illustration of the fundamental theory while analysing dynamical systems and validating the chaotic behaviour of the Lorenz–Haken (LH) equations, a system of fractional differential equations. The LH equations are used to describe the 4D chaotic laser. The Adams–Bashforth numerical method is used to extract the numerical solutions projected model. The classical model introduces the bifurcation of the parameter linked with the system. The system's uniqueness and existence are confirmed using the fixed-point hypothesis with the Caputo–Fabrizio fractional operator, followed by boundedness and dynamical analysis. Furthermore, the chaotic character of the numerical solutions with different orders obtained at different beginning circumstances is presented. [ABSTRACT FROM AUTHOR]

The modified exponential function method is used to obtain some new analytical solutions of the nonlinear Gerdjikov–Ivanov equation with the M-fractional operator. The novel obtained solutions are expressed in hyperbolic, trigonometric, and exponential function forms. Moreover, the instability modulation and gain spectra of the Gerdjikov–Ivanov equation are also analyzed. Constraints conditions are utilized to verify the existence of the solutions. Presented solutions are novel, satisfy and verify the M-fractional Gerdjikov–Ivanov equation. [ABSTRACT FROM AUTHOR]

This article deals with the applications of (1 / G ′) -expansion method which has recently been presented to the literature. Applying this tool on the generalized (2 + 1)-dimensional Hirota–Maccari system, perturbed nonlinear Schrödinger's equation and dispersive cubic-quintic nonlinear Schrödinger equation, new complex traveling wave solutions are founded. The solution of partial differential equations representing the phenomenon of fiber optic communication and the interrelationship of the internal dynamics of traveling wave solutions, which play an important role in the transport of energy, have been discussed. Furthermore, for a better physical understanding of the results, various graphs with the appropriate values of the parameters have been presented and the simulation has been used to support the discussion. [ABSTRACT FROM AUTHOR]

This research manuscript focuses on the extraction of dark-bright solitons and periodic wave distributions of two models, namely, the Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and complex Kundu–Eckhaus equation with conformable derivative. To reach these important properties, the generalized exponential rational function method is considered. To observe wave distributions in periodic and singular sense, dynamical behaviour modulus of solutions are also archived. Strain conditions for validity of results obtained in this paper are also reported. [ABSTRACT FROM AUTHOR]

In this paper, perturbed Chen–Lee–Liu equation is considered to describe pulse propagation in optical fibers. Chen–Lee–Liu equation is a derivative form of the nonlinear Schrödinger equation. Two renowned analytical techniques namely, exp (- φ (η)) -expansion method and the generalized Kudryashov method are applied to analyze this model. The algorithm of these methods is also presented. Different structures of soliton solutions are successfully investigated for perturbed this model. Additionally, constraint conditions on coefficients of perturbation terms are also defined to assure the existence of such solitons. The graphical representation of these solutions is shown to demonstrate the dynamics of pulse propagation governed by Chen–Lee–Liu equation in optical fibers. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the higher-order dispersive extended nonlinear Schrödinger equation, which demonstrates the propagation of ultrashort pulses in optical communication networks. In this study, both the sinh-Gordon expansion method and the generalized exponential rational function method are used to offer some novel optical solutions. These optical soliton solutions are dark soliton, bright soliton, singular, periodic, and dark-bright soliton solutions. The obtained optical soliton solutions are presented graphically in 2D and 3D to clarify the behavior of solutions more effectively. The constraint conditions are also used to verify the exitances of the new analytical solutions. Moreover, all solutions compared to solutions obtained previously are new, and all the new wave solutions have verified Eq. (1) after we substituted them into the studied equation. In the future, these novel soliton solutions will be very helpful in developing fluid dynamics, biomedical issues, dynamics of adiabatic parameters, industrial research, and many other areas of science. To our acknowledgment, the presented optical solutions are novel, and also beforehand these methods have not been applied to this studied equation. [ABSTRACT FROM AUTHOR]

In this paper, we use the homotopy perturbation transform method (HPTM) to offer an efficient semi-analytical technique for solving fractional Emden–Fowler equations. A mixture of Laplace transform, Caputo–Fabrizio derivative, and homotopy perturbation transformation process has the projected technique. To assess the efficacy of the suggested technique, test examples have been provided. The series have been used to represent semi-analytical solutions. Also, covered have the convergence position, estimation, and semi-analytical simulation results. The HPTM efficiently managed and controlled a series solution that quickly converges to a precise result in a narrow admissible region. The new findings essentially improve and simplify some of the previously published findings (see Malagia in Math. Comput. Simul. 190:362, 2021). By assigning appropriate values to free parameters, dynamical wave structures of some semi-analytical solutions are graphically demonstrated using 2-dimensional and 3-dimensional figures. Furthermore, various simulations are used to demonstrate the physical behaviors of the acquired solution with respect to fractional integer order. [ABSTRACT FROM AUTHOR]

In this work, two new adaptations for the trigonometric and hyperbolic trigonometric function approaches have been presented. These two modifications, entitled modified extended rational sin – cos function technique and modified extended rational sinh – cosh function method, have been applied for the first time to the Fokas system that represents the nonlinear pulse propagation in monomode fiber optics. We intend to produce innovative, explicit traveling waves, solitons, and periodic wave solutions. These achieved outcomes are presented in the form of exponential functions, trigonometric hyperbolic functions, and combination constructions of the exponential functions along with the trigonometric and hyperbolic trigonometric functions. The obtained solutions reveal significant features of the physical phenomenon and are new. The investigated model incorporates the notions of dispersion, transverse diffusion, degree of dispersion, nonlinear pairing, nonlinear immersion, and the force of the nonlinear interaction among the two components of the system. For the most accurate visual evaluation of the physical importance and dynamic properties, we have presented the findings in a variety of plots, which involve two- and three-dimensional representations. One or more elements in our research that are unique, such as newly modified methodologies, is a new observation that leads researchers to invest in new solutions. [ABSTRACT FROM AUTHOR]

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α ∈ (0 , 1 ] is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof. [ABSTRACT FROM AUTHOR]

The thermophoretic motion (TM) system with a variable heat transmission factor, based on the Korteweg-de Vries (KdV) equation, is used to model soliton-like thermophoresis of creases in graphene sheets. Painlevé test is employed to discover that the equation is Painlevé integrable. Then an auto-Bäcklund transformation using the truncated Painlevé expansion is obtained. Concerning the additional variables, the auto-Bäcklund transformations convert the nonlinear model to a set of linear partial differential equations. Finally, various explicit precise solutions based on the acquired auto-Bäcklund transformations are investigated and the researched solutions are illustrated in 3D, 2D and contour plots. Furthermore, the Cole-Hopf transformation is used in conjunction with Hirota's bilinear technique to get multiple regular and singular kink solutions. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear Schrödinger equation in the sense of conformable derivative. Bifurcations and phase portraits are first proposed by using bifurcation theory, which investigates the dynamical behavior of this equation. This bifurcation theory classifies the plausible solutions to infinite periodic wave solutions, periodic wave solutions, two kink (anti-kink) wave solutions, and two families of breaking wave solutions. A generalized double auxiliary equation approach that generates three families of exact exact traveling wave solutions is then proposed using the conformable operator under various parameter conditions. The 3D behavior of various solutions with absolute real and imaginary parts is displayed. The obtained results show that the proposed methodology is efficient and applicable to a broad class of conformable nonlinear partial differential equations in mathematical physics. [ABSTRACT FROM AUTHOR]

The aim of this article is to explore the characteristics of fluid flow using the process of melting and entropy generation past on the Riga plate. Cross fluid is competent to explore the characteristics of fluid with high and low deformations under the influenced force. Transport of heat mechanism is inspected with key facts about melting conditions, viscous dissipation and thermal radiation. For the inspection of electromagnetic hydrodynamic aspects of fluid, Riga geometry due to this Lorentz force is produced. The partial differential form of the equations has been converted into an ordinary differential system with the use of similarity transformations. The reliable numerical Keller Box scheme is being applied to solve the obtained system of ordinary equations and comparison of the results have been performed through the built-in Matlab solver bvp4c. The obtained solutions have also been compared with the literature results. The physical quantities and their numerical attitude are provided in the form of statistical graphs. The Lorentz forces produce to slow down the fluid's velocity, while the Hartman number accelerated the velocity. [ABSTRACT FROM AUTHOR]

This work explores the magneto-hydrodynamics (MHD) Jeffery–Hamel nanofluid flow between two rigid non-parallel plane walls with heat transfer by employing hybrid nanoparticles, especially Cu and Cu-Al 2 O 3 . Here the MHD nanofluid flow problem is extended with fuzzy volume fraction and heat transfer with diverse nanoparticles to cover the influence of thermal profiles with hybrid nanoparticles on the fuzzy velocity profiles. The nanoparticle volume fraction is described with a triangular fuzzy number ranging from 0 to 5 % . A novel double parametric form-based homotopy analysis approach is considered to study the fuzzy velocity and temperature profiles with hybrid nanoparticles in both convergent and divergent channel positions. Finally, the efficiency of the proposed method has been demonstrated by comparing it with the available results in a crisp environment for validation. [ABSTRACT FROM AUTHOR]