Generalized Riccati Equation Research Articles

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.

New Dynamic Multiwave Solutions of the Fractional Peyrard–Bishop DNA Model

Abstract In this paper, we have studied the new solitary wave solutions of the beta-fractional derivative form of the Peyrard–Bishop DNA model (PB-DNAM). These solutions are responsible for analyzing the nonlinear interaction between the adjacent displacements of the DNA strand. To get these solutions, we have applied the generalized Riccati equation expansion method. Under different parametric conditions and fractional values, the obtained solutions show different wave patterns including w-shape, bright, combined dark-bright, periodic wave solutions, bell shape, m-shape, w-shape along with two bright solutions, and m-shape along with two dark solutions. These physical characteristics are analyzed thoroughly by graphical representations. The solutions show the successful application of the proposed method which will be helpful in finding analytical solutions to other nonlinear problems.

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.

Revisiting (2+1)-dimensional Burgers’ dynamical equations: analytical approach and Reynolds number examination

Classical Burgers’ equation is an indispensable dynamical evolution equation that is autonomously devised by Burgers and Harry Bateman in 1915 and 1948, respectively. This important model is featured through a nonlinear partial differential equation (NPDE). Furthermore, the model plays a crucial role in many areas of mathematical physics, including, for instance, fluid dynamics, traffic flow, nonlinear acoustics, turbulence phenomena, and linking convection and diffusion processes to state a few. Thus, in the present study, an efficient analytical approach by the name ‘generalized Riccati equation approach’ is adopted to securitize the class of (2+1)-dimensional Burgers’ equations by revealing yet another set of analytical structures to the governing single and vector-coupled Burgers’ equations. In fact, the besieged method of the solution has been proven to divulge various sets of hyperbolic, periodic, and other forms of exact solutions. Moreover, the method first begins by transforming the targeted NPDE to a nonlinear ordinary differential equation (NODE), and subsequently to a set of an algebraic system of equations; where the algebraic system is then solved simultaneously to obtain the solution possibilities. Lastly, certain graphical illustrations in 2- and 3-dimensional plots are set to be depicted - featuring the evolutional nature of the resulting structures, and thereafter, analyze the influence of the Reynolds number Ra on the respective wave profiles.

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

An addendum to the problem of zero-sum LQ stochastic mean-field dynamic games

In this paper, we first address a linear quadratic mean-field game problem with a leader–follower structure. By adopting a Riccati-type approach, we show how one can obtain a state-feedback representation of the pairs of strategies which achieve an open-loop Stackelberg equilibrium in terms of the global solutions of a system of coupled matrix differential Riccati-type equations. In the second part of this paper, we obtain necessary and sufficient conditions for the solvability of the involved coupled generalized Riccati equations.

Optical solitons of the perturbed nonlinear Schrödinger equation using Lie symmetry method

This paper presents an analytical treatment of the perturbed nonlinear Schrödinger equation (P-NLSE) using the Lie symmetry method. The NLSE is a fundamental equation in nonlinear optics and describes the propagation of optical pulses in a nonlinear medium. In the presence of perturbations, the solution to the NLSE becomes more complicated and requires more sophisticated mathematical techniques to analyze. The Lie symmetry method is a powerful tool for finding exact solutions to nonlinear differential equations and has been applied successfully to a wide range of problems in physics and engineering. In this paper, we use the Lie symmetry method to obtain exact analytical solutions for the P-NLSE and study their properties. Our results show that the perturbations can have a significant effect on the behavior of the optical pulses, leading to phenomena such as soliton fission and dispersive shock waves. The Lie symmetry method is used to derive three generators for the Lie algebra, and two reductions are examined through combinations of vector generators. The Improved generalized Riccati equation and Nucci reduction method are then employed to obtain exact solutions, including bright soliton, multiple singular soliton, and exponential solutions.

Construction of Infinite Series Exact Solitary Wave Solution of the KPI Equation via an Auxiliary Equation Method

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models.

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.

Optical solitons of the perturbation Fokas–Lenells equation by two different integration procedures

In this study, the general projective Riccati equations and the enhanced Kudryashov’s methods are presented for the optical solitons of the perturbation Fokas–Lenells (FL) equation. For this aim, the perturbation Fokas–Lenells equation is converted to a nonlinear ordinary differential equation via wave transformations. Then, the balance number is calculated by the homogeneous balance method. Then, the used methods are applied to the given model. Finally, three-dimensional, contour, and two-dimensional figures of the results are presented. According to the obtained results, both methods are effective and powerful. The obtained results are significant in mathematical physics, soliton theory, and many other areas.

On the Basic Theory of Some Generalized and Fractional Derivatives

We continue the development of the basic theory of generalized derivatives as introduced and give some of their applications. In particular, we formulate necessary conditions for extrema, Rolle’s theorem, the mean value theorem, the fundamental theorem of calculus, integration by parts, along with an existence and uniqueness theorem for a generalized Riccati equation, each of which provides simple proofs of the corresponding version for the so-called conformable fractional derivatives considered by many. Finally, we show that for each α>1 there is a fractional derivative and a corresponding function whose fractional derivative fails to exist everywhere on the real line.

Optical solitons to the fractional order nonlinear complex model for wave packet envelope

This research deals with the fractional order nonlinear complex model, which is a general form of nonlinear Schrodinger equation. This model demonstrates boson gas among 2 as well as 3 body collisions, nuclear hydro-dynamics allied with Skyrme energies, in addition the optical rhythm propagations within dielectric non-Kerr media. To analyze and generate complex dynamics, optical soliton solutions of the model are derived through improved generalized Riccati equation mapping scheme. As a result, bright, dark and combo bright-dark optical wave envelopes and oscillating pulse bright optical solitons are obtained. The effects of fractionality and parameters are also illustrated with few numerical graphics of the achieved results. The achieved optical (bright and dark) solitons solutions can be used in signal transmission through optical fiber communications system.

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.

Linear Quadratic Optimal Control Problem for Linear Stochastic Generalized System in Hilbert Spaces

A finite-horizon linear stochastic quadratic optimal control problem is investigated by the GE-evolution operator in the sense of the mild solution in Hilbert spaces. We assume that the coefficient operator of the differential term is a bounded linear operator and that the state and input operators are time-varying in the dynamic equation of the problem. Optimal state feedback along with the well-posedness of the generalized Riccati equation is obtained for the finite-horizon case. The results are also applicable to the linear quadratic optimal control problem of ordinary time-varying linear stochastic systems.

New Solitary-Wave Solutions of the Van der Waals Normal Form for Granular Materials via New Auxiliary Equation Method

In this paper, we use general Riccati equation to construct new solitary wave solutions of the Van der Waals normal form, which is one of the most famous models for natural and industrial granular materials. It is very important to understand the static and dynamic characteristics of this models in many application fields. We solve the general Riccati equation through different function transformation, and many new hyperbolic function solutions are obtained. Then, it is substituted into the Van der Waals normal form as an auxiliary equation. Abundant types of solitary-wave solutions are obtained by choosing different coefficient in the general Riccati equation, and some of them have not been found in other documents. The results show that the analysis method we used is very simple and effective for dealing with nonlinear models.

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

Abundant closed-form solutions of the (3+1)-dimensional Vakhnenko-Parkes equation describing the dynamics of various solitary waves in ocean engineering

This research article is a study of the (3+1)-dimensional Vakhnenko-Parkes equation that governs the propagation of high-frequency waves in a relaxing medium. To find a variety of new analytic exact solitary wave solutions, we apply three effective schemes, such as the generalized Kudryashov method (GKM), the generalized exponential rational function method (GERFM), and the generalized Riccati equation mapping method (GREMM). Obtained solutions are crucial to understand some existing, complicated physical phenomena. Many solutions are obtained, which are expressed by the exponential, trigonometric, and hyperbolic functions, including tanh, coth, sech, csch, tan, cot, sech, csch, and their combinations. These solutions include periodic-wave solutions, kink waves, combinations of kink and multi solitons, singular solitons, singular-kink structures, and periodic solitons. The 3D and 2D plots are shown by considering the possibility of arbitrary parameters occurring in the solutions better to understand the underlying mechanisms of the stated family. The computational results of this study assure us that the proposed approaches seem useful, efficient, and reliable tools for estimating the solutions attained in the study. Moreover, the method is expected to be frequently applied to examine various types of nonlinear PDEs, which emerge in mathematical physics, ocean engineering, and science.

An infinite‐dimensional affine stochastic volatility model

AbstractWe introduce a flexible and tractable infinite‐dimensional stochastic volatility model. More specifically, we consider a Hilbert space valued Ornstein–Uhlenbeck‐type process, whose instantaneous covariance is given by a pure‐jump stochastic process taking values in the cone of positive self‐adjoint Hilbert–Schmidt operators. The tractability of our model lies in the fact that the two processes involved are jointly affine, that is, we show that their characteristic function can be given explicitly in terms of the solutions to a set of generalized Riccati equations. The flexibility lies in the fact that we allow multiple modeling options for the instantaneous covariance process, including state‐dependent jump intensity. Infinite dimensional volatility models arise, for example, when considering the dynamics of forward rate functions in the Heath–Jarrow–Morton–Musiela (HJMM) modeling framework using the Filipović space. In this setting, we discuss various examples: an infinite‐dimensional version of the Barndorf–Nielsen–Shephard stochastic volatility model, as well as covariance processes with a state dependent intensity.