Riccati Equation Mapping Method Research Articles

Exact solutions and bifurcations of the new (3+1)-dimensional generalized Kadomstev-Petviashvili equation

Abstract The exact solutions of the new generalized (3+1)-dimensional Kadomtsev-Petviashvili (ngKP) equation by using the modified extended tanh-function method, the improved (G′/G) method and the generalized Riccati equation mapping method are considered. Different types of exact traveling wave solutions are obtained, enriching the exact solutions of this equation. In addition, the dynamical behaviors of the different types of solutions obtained by the equation are graphically presented using 3D, 2D and density plots. Then, the phase portraits of the equations are obtained through the bifurcation theory to verify the validity of the previous three methods. Finally, the comparison between our results and the known results is given, it can be concluded that we have provided a large number of new travelling wave solutions to the equation.

On the study of solitary wave dynamics and interaction phenomena in the ultrasound imaging modelled by the fractional nonlinear system

This research work focuses on investigating the propagation of ultrasonic waves, which propagate mechanical vibrations of molecules or particles inside materials. Ultrasound imaging is extensively used and deeply rooted in the medical field. The key technologies that form the basis for many different uses in the area include transducers, contrast agents, pulse compression, beam shaping, tissue harmonic imaging, techniques for measuring blood flow and tissue motion, and three-dimensional imaging. The third-order non-linear β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}-fractional Westervelt model has been used as a governing model in the imaging process for securing the different wave structures. The exact solutions of different types, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted. These solutions are obtained by using two newly introduced techniques, namely modified generalized Riccati equation mapping method and modified generalized exponential rational function method. Moreover, breather, lump and other waves are extracted by the assistance of logarithmic transformation and different test functions. The used methodologies are extremely effective and possess substantial computing capacity to effectively address the different solutions with a high level of accuracy in these systems. The techniques used are well-known for being effective, simple, and flexible enough to integrate multiple soliton systems into a unified framework. In addition, we provide 2D and 3D graphs that explain the behavior of the solution at various parameter values, under the influence of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}-fractional derivatives. The results offered in this study may improve the comprehension of the nonlinear dynamic behavior of the specific system and confirm the efficacy of the approaches used. We expect that our approaches will be beneficial for a wide range of nonlinear models and other problems in the related fields.

On the study of double dispersive equation in the Murnaghan’s rod: Dynamics of diversity wave structures

This article secures the various wave structures of the fractional double dispersive equation, a significant nonlinear equation that describes the propagation of nonlinear waves within the elastic, uniform, and inhomogeneous Murnaghan’s rod. The model under discussion has a wide range of applications in science and engineering. Two recently developed analytical techniques known as the improved generalized Riccati equation mapping method and the multivariate generalized exponential rational integral function method have been applied to the proposed equation for the first time. A variety of solutions have been revealed such that dark, singular, bright-dark, bright, complex, and combined solitons. Furthermore, we include a diverse array of plots that illustrate the physical interpretation of the obtained solutions in relation to a number of significant parameters, thereby highlighting the impact of fractional derivatives. Within the context of the proposed model, these visualizations give a clear understanding of the behavior and characteristics of the solutions. This study’s results have the potential to enhance comprehension of the nonlinear dynamic characteristics exhibited by the specified system and validate the efficacy of the implemented techniques. The achieved results significantly enhance our understanding of nonlinear science and the nonlinear wave fields associated with more complex nonlinear models.

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Unveiling new insights into soliton solutions and sensitivity analysis of the Shynaray-IIA equation through improved generalized Riccati equation mapping method

Unveiling new insights into soliton solutions and sensitivity analysis of the Shynaray-IIA equation through improved generalized Riccati equation mapping method

Optical soliton solutions in a distinctive class of nonlinear Schrödinger’s equation with cubic, quintic, septic, and nonic nonlinearities

The Biswas–Mollivic equation is a special type of nonlinear Schrödinger equation, which explains the spatio-temporal behaviour of excitable media. In this paper, we investigate the optical soliton solutions of the Biswas–Mollivic equation with cubic–quintic–septic–nonic nonlinearities using the generalized Riccati equation mapping method. This method is efficient and provides new perspectives. It also provides novel insights into the dynamics of excitable media. Our findings add to a better understanding of the complex spatio-temporal patterns that develop in excitable media and have potential applications in the design of new technologies for controlling and manipulating pattern formation. To depict optical soliton solutions graphically, we use the MATLAB software.

Investigation of Brownian motion in stochastic Schrödinger wave equation using the modified generalized Riccati equation mapping method

Investigation of Brownian motion in stochastic Schrödinger wave equation using the modified generalized Riccati equation mapping method

Comparisons of Numerical and Solitary Wave Solutions for the Stochastic Reaction–Diffusion Biofilm Model including Quorum Sensing

This study deals with a stochastic reaction–diffusion biofilm model under quorum sensing. Quorum sensing is a process of communication between cells that permits bacterial communication about cell density and alterations in gene expression. This model produces two results: the bacterial concentration, which over time demonstrates the development and decomposition of the biofilm, and the biofilm bacteria collaboration, which demonstrates the potency of resistance and defense against environmental stimuli. In this study, we investigate numerical solutions and exact solitary wave solutions with the presence of randomness. The finite difference scheme is proposed for the sake of numerical solutions while the generalized Riccati equation mapping method is applied to construct exact solitary wave solutions. The numerical scheme is analyzed by checking consistency and stability. The consistency of the scheme is gained under the mean square sense while the stability condition is gained by the help of the Von Neumann criteria. Exact stochastic solitary wave solutions are constructed in the form of hyperbolic, trigonometric, and rational forms. Some solutions are plots in 3D and 2D form to show dark, bright and solitary wave solutions and the effects of noise as well. Mainly, the numerical results are compared with the exact solitary wave solutions with the help of unique physical problems. The comparison plots are dispatched in three dimensions and line representations as well as by selecting different values of parameters.

Investigation of soliton structures for dispersion, dissipation, and reaction time-fractional KdV–burgers–Fisher equation with the noise effect

ABSTRACT In this manuscript, the soliton structures for the time-fractional KdV–Burgers–Fisher equation with the effect of noise are investigated analytically. This is the dispersion–dissipation–reaction model. The third- and fifth-order time-fractional stochastic KdV–Burgers–Fisher equations are under consideration. These wave structures are constructed with the help of an extended generalized Riccati equation mapping method (EGREM). This method is a combined form of the G ′ / G expansion method with the generalized Riccati equation mapping method, and it will give the different forms of wave structures like shock, singular, combo, hyperbolic, trigonometric, mixed trigonometric, and rational solutions. These techniques are used symbolically with computational tools like Mathematica to demonstrate the efficiency and simplicity of the proposed strategy. Additionally, with the various relevant parameter values, the sketches of some solutions in the form of 3D and contour representations for the purpose of comprehending physical processes are drawn. These sketches clearly show the random behavior of these wave structures that are appearing in dispersion, dissipation, and reaction concentrations of these mathematical models.

Analytical study of reaction diffusion Lengyel-Epstein system by generalized Riccati equation mapping method

In this study, the Lengyel-Epstein system is under investigation analytically. This is the reaction–diffusion system leading to the concentration of the inhibitor chlorite and the activator iodide, respectively. These concentrations of the inhibitor chlorite and the activator iodide are shown in the form of wave solutions. This is a reaction†“diffusion model which considered for the first time analytically to explore the different abundant families of solitary wave structures. These exact solitary wave solutions are obtained by applying the generalized Riccati equation mapping method. The single and combined wave solutions are observed in shock, complex solitary-shock, shock singular, and periodic-singular forms. The rational solutions also emerged during the derivation. In the Lengyel-Epstein system, solitary waves can propagate at various rates. The harmony of the system’s diffusive and reactive effects frequently governs the speed of a single wave. Solitary waves can move at a variety of speeds depending on the factors and reaction kinetics. To show their physical behavior, the 3D and their corresponding contour plots are drawn for the different values of constants.

Stability analysis and consistent solitary wave solutions for the reaction–diffusion regularized nonlinear model

The reaction-nonlinear diffusion model is analyze analytically, under both logistic and bistable growth regularization. These regularization techniques have been widely apply and employ in models of chemical phase separation. In this study, we mainly focus on construct the different wave structures are with the help of the generalized Riccati equation mapping method. This method is provided us the different hyperbolic, trigonometric, and rational solutions. These solutions help for the dynamic study of the reaction-nonlinear diffusion model. Moreover, for the physical significance, we construct the 3D and their corresponding contour by choosing the different values of the parameters. Additionally for the computational interests of the readers stability and consistency analysis were proved.

Sensitivity analysis and solitary wave solutions to the (2+1)-dimensional Boussinesq equation in dispersive media

This paper explores the dynamic behavior of the [Formula: see text]-dimensional Boussinesq equation, which is a nonlinear water wave equation used to model wave packets in dispersive media with weak nonlinearity. Specifically, we investigate the equation’s traveling wave solutions using the Riccati equation mapping method. Our results include solitary and soliton solutions, each with their own set of parameter values. To provide a comprehensive understanding of these solutions, we present them in general form and visualize their significance using various graphs, such as 3D, 2D, and contour plots. The computational effort and resulting outcomes highlight the efficacy of our approach, which has the potential to be applied to other nonlinear physical problems in fields such as mathematical physics, engineering, and nonlinear science.

Optical fiber application of the Improved Generalized Riccati Equation Mapping method to the perturbed nonlinear Chen-Lee-Liu dynamical equation

This work considers the improved generalized Riccati equation mapping (IGREM) technique with its Bäcklund transformation to obtain the wave solutions of the perturbed Chen-Lee-Liu (CLL) equation. The CLL equation actually depicts pulse propagation in optical fibers and plasma. We utilize proposed technique to obtain hyperbolic, periodic trigonometric, rational and exponential function solutions. To make this work worthy and for efficient illustration of this methodology, 3D, contours and 2D plots of some selected solutions are constructed by selecting the values of involved parameters. The results of this paper prove that the used method is effective to improve the nonlinear dynamical behavior of a system.

Analytical soliton solutions and wave profiles of the (3+1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation

In this research paper, the improved generalized Riccati equation mapping (IGREM) method is employed to get analytical soliton solutions of (3+1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV-ZK) equation. By giving appropriate parametric values, we obtain soliton solutions of various types, such as kink soliton, singular soliton, and periodic-singular soliton. The description provided in physical variables allows for the study of genuine multispecies plasmas, plasma models, and frequency regimes. These solutions are essential in illuminating several physical phenomena in engineering and other applied sciences. In addition, 3D and 2D graphs of some selected solutions are sketched for the best physical characterization of the obtained results. This method can also be implemented in many non-linear equations in contemporary research areas. Comparison with previous studies shows that the discovery of singular soliton is novel. Therefore, our research represents a significant advancement in the field and contributes to expanding the knowledge regarding soliton behavior. The applications of this study include electron and ion-acoustic modes in regular plasmas with hot Boltzmann electron species, as well as ion and dust-acoustic modes, depending on the specific modeling of the heavier components.

A study of propagation of the ultra-short femtosecond pulses in an optical fiber by using the extended generalized Riccati equation mapping method

A study of propagation of the ultra-short femtosecond pulses in an optical fiber by using the extended generalized Riccati equation mapping method

On the Dynamics of Solitary Waves to a (3+1)-Dimensional Stochastic Boiti–Leon–Manna–Pempinelli Model in Incompressible Fluid

We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He’s semi-inverse method and the Riccati equation mapping method, we can acquire the solutions of the SBLMPE. Since the Boiti–Leon–Manna–Pempinelli equation is utilized to explain incompressible liquid in fluid mechanics, the acquired solutions may be applied to explain a lot of fascinating physical phenomena. To address how Brownian motion effects the exact solutions of the SBLMPE, we present some 2D and 3D diagrams.

New exact solutions to space–time fractional telegraph equation with conformable derivative

In this paper, we extract variety of new exact traveling wave solutions of space–time fractional nonlinear Telegraph equation for transmission lines by using improved generalized Riccati equation mapping (IGREM) method. The aforementioned equation has been solved for the first time using conformable fractional derivative. The nonlinear Telegraph equation plays crucial role in the mathematical study of voltage and current on an electrical transmission line. In an effort to motivate the researchers, we have used very efficacious technique called IGREM method to find a variety of soliton solutions, in which appropriate conformable fractional wave transformation converts nonlinear partial fractional differential equation into nonlinear ordinary equation which is then solved to find the exact solutions of equation. Hence, we succeed in constructing rich solutions that include periodic wave, singular periodic wave solutions, dark, combined dark–bright and rational solutions. To investigate wave profiles of various solutions, 2D and 3D graphs have been explained. The originality of this work has been proved by comparing the obtained results with previously present results in literature. This method has substantial advantage over other methods that it can be used to solve higher order nonlinear PDEs, and this method allows to solve these equations more conveniently with the aid of mathematical software.

Comparative analysis of numerical and newly constructed soliton solutions of stochastic Fisher-type equations in a sufficiently long habitat

This paper is a key contribution with respect to the applications of solitary wave solutions to the unique solution in the presence of the auxiliary data. Hence, this study provides an insight for the unique selection of solitons for the physical problems. Additionally, the novel numerical scheme is developed to compare the result. Further, this paper deals with the stochastic Fisher-type equation numerically and analytically with a time noise process. The nonstandard finite difference scheme of stochastic Fisher-type equation is proposed. The stability analysis and consistency of this proposed scheme are constructed with the help of Von Neumann analysis and Itô integral. This model is applicable in the wave proliferation of a viral mutant in an infinitely long habitat. Additionally, for the sake of exact solutions, we used the Riccati equation mapping method. The solutions are constructed in the form of hyperbolic, trigonometric and rational forms with the help of Mathematica 11.1. Lastly, the graphical comparisons of numerical solutions with exact wave solution with the help of Neumann boundary conditions are constructed successfully in the form of 3D and line graphs by using different values of parameters.

Construction of Exact Solutions for Gilson–Pickering Model Using Two Different Approaches

In this paper, the extended simple equation method (ESEM) and the generalized Riccati equation mapping (GREM) method are applied to the nonlinear third-order Gilson–Pickering (GP) model to obtain a variety of new exact wave solutions. With the suitable selection of parameters involved in the model, some familiar physical governing models such as the Camassa–Holm (CH) equation, the Fornberg–Whitham (FW) equation, and the Rosenau–Hyman (RH) equation are obtained. The graphical representation of solutions under different constraints shows the dark, bright, combined dark–bright, periodic, singular, and kink soliton. For the graphical representation, 3D plots, contour plots, and 2D plots of some acquired solutions are illustrated. The obtained wave solutions motivate researchers to enhance their theories to the best of their capacities and to utilize the outcomes in other nonlinear cases. The executed methods are shown to be practical and straightforward for approaching the considered equation and may be utilized to study abundant types of NLEEs arising in physics, engineering, and applied sciences.

Dynamical behaviors and abundant optical soliton solutions of the cold bosonic atoms in a zig-zag optical lattice model using two integral schemes

In this present paper, we obtain hyperbolic, exponential, trigonometric function, other soliton solutions, and their combinations for the cold bosonic atoms in a zig-zag optical lattice model based on two efficient methods, such as the generalized Riccati equation mapping (GREM) method and generalized Kudryashov (GK) method. The used techniques are very reliable and effective tools and provide numerous exact soliton solutions of the nonlinear PDE. The zig-zag optical lattice model, widely used to represent the nonlinear wave and the soliton dynamics in fluid dynamics and plasma physics, is examined in this article to obtain exact optical soliton solutions and study their physical properties. For this, we first convert a partial differential equation (PDE) into an ordinary differential equation (ODE) by employing wave transformation and then split the equation into imaginary and real parts. The derived optical soliton solutions are illustrated graphically using Mathematica software to distinguish constant parameter values. Consequently, bell-shape, anti-bell-shape, traveling wave, periodic, mix periodic, singular soliton, and some new types of solitons demonstrate to validate these acquired outcomes with physical phenomena and make the results worthy. Furthermore, the 3D, 2D, and contour graphs are sketched to assign suitable constant parameters to illustrate the physical phenomena of the obtained solutions. The accomplished soliton solutions indicate that the applied computational system is a direct, reliable, productive, and more complex physical phenomenon. Symbolic computation is used in the software package Mathematica to obtain the various soliton solutions and different dynamical behavior of the newly formed solutions