Maccari System Research Articles

On exact solutions of the (2+1)-dimensional time conformable Maccari system

For the ([Formula: see text])-dimensional time conformable Maccari system with beta-derivative, exact solutions are obtained in this paper using the improved sub-equation method. This technique has been used to find rational solutions, generalized hyperbolic function solutions and generalized trigonometric function solutions. The new structures of the solutions are shown after the proper parameter values have been given. It is easier to understand the intricate physical processes of these dynamical models by viewing the physical surfaces of some obtained solutions in various forms. The outcomes demonstrate the efficiency of the applied strategy, which may be used to solve a number of nonlinear physical problems. ([Formula: see text])-dimensional time conformable Maccari system is considered and the graphs of solutions are sketched using MAPLE software after its exact solutions which are acquired using the improved sub-equation method.

The mathematical and wave profile analysis of the Maccari system in nonlinear physical phenomena

The mathematical and wave profile analysis of the Maccari system in nonlinear physical phenomena

Exact solutions of the stochastic Maccari system forced by multiplicative noise

AbstractIn the Itô sense, we look at the stochastic Maccari system (SMS) with multiplicative noise. To generate new analytical stochastic solutions, we use three separate techniques including sine‐cosine, Riccati‐Bernoulli sub‐ODE, and semi‐inverse. Such solutions are critical for comprehending some fascinating and complex physical phenomena. In addition, some results from earlier studies are generalized. Furthermore, we investigate the impact of stochastic term on the exact solutions of the SMS.

Two Analytical Schemes for the Optical Soliton Solution of the (2 + 1) Hirota–Maccari System Observed in Single-Mode Fibers

In this scientific research article, the new Kudryashov method and the tanh-coth method, which have not been applied before, are employed to construct analytical and soliton solutions of the (2+1)-dimensional Hirota–Maccari system. The (2+1)-dimensional Hirota–Maccari system is a special kind of nonlinear Schrödinger equation (NLSEs) that models the motion of isolated waves localized in a small part of space, and is used in such various fields as fiber optics telecommunication systems, nonlinear optics, plasma physics, and hydrodynamics. In addition, the Hirota–Maccari system defines the dynamical characters of femtosecond soliton pulse propagation in single-mode fibers. Analytical solutions of the model are successfully acquired with the assistance of symbolic computation utilizing these methods. Finally, 3D, 2D, and contour graphs of solutions are depicted at specific values of parameters. It is shown that the new Kudryashov method and the tanh-coth method are uncomplicated, very effective, easily applicable, reliable, and indeed vital mathematical tools in solving nonlinear models.

Rogue Waves Generator and Chaotic and Fractal Behavior of the Maccari System with a Resonant Parametric Forcing

Using the Asymptotic Perturbation (AP) method we can find approximate solutions for the Maccari equation with a parametric resonant forcing acting over the frequency of a generic mode. Taking into account its nonlocal behavior and applying symmetry considerations, a system with two coupled equations for the phase and amplitude modulation can be obtained. The system can be solved, and we demonstrate the existence of a big modulation in the wave amplitude, producing a rogue waves train and, in this case, these waves are not isolated. We then obtain a rogue waves generator, being able of producing and controlling the rogue waves’ amplitude. Another important finding is the existence of chaotic or fractal solutions, because of the presence of an arbitrary function in the solution.

Exploring the fractional Hirota Maccari system for its soliton solutions via impressive analytical strategies

In the field of marine engineering, the characteristics of wave propagation play an imperative character. In many geographical regions, the key source of environmental effects on artificial floating or stationary structures or seashores is waves. This article deals with the Hirota Maccari system (HMs) via a new fractional derivative operator known as the beta derivative. This aforesaid equation is a significant model which deals with a variety of nonlinear phenomenons in the fields of optical fibers, physics and other scientific fields. A complex wave hypothesis is applied to transform the aforesaid system with beta derivative into an ODE system. The modified Kudryashov’s (New version) and the Auxiliary equation methods are used to build a variety of soliton solutions of HMs with beta derivative. The singular, periodic, bright, dark solitons and mixed solitons are constructed with the above-mentioned approaches via soft computation in Mathematica. Some of them are numerically simulated for 2D and 3D representation. All the solutions of fractional HMs produced by the above two techniques are novel and have not been derived yet.

New interaction of high-order breather, periodic-wave, lump, rational soliton solutions and mixed solutions for the Hirota–Maccari system

The higher-order breather, periodic-wave, lump, rational soliton solutions and mixed solutions of the Hirota–Maccari (HM) system by virtue of the Kadomtsev–Petviashvili (KP) hierarchy reduction method are investigated in this work. Through analyzing the structural characteristics of periodic-wave solutions, we attain the quasi-periodic W(M)-shaped waves and two kinds of breathers. The mixed solutions that consist of the quasi-periodic W(M)-shaped waves and breathers are constructed. Further, by taking the long wave limit on the periodic-wave solutions, the semi-rational solutions are derived, which illustrate the interaction of the rational soliton, lump, quasi-periodic wave and breather. Characteristics of these mixed solutions are discussed graphically and the corresponding generating conditions are given. Especially, a new bound-state interaction composed of lump and breather is generated under the velocity resonance mechanism. This newfangled pattern is a beautiful phenomenon for the HM system.

Rational and Semi-Rational Solutions to the (2 + 1)-Dimensional Maccari System

The KP hierarchy reduction method is one of the most reliable and efficient techniques for determining exact solitary wave solutions to nonlinear partial differential equations. In this paper, according to the KP hierarchy reduction technique, rational and some other semi-rational solutions to the (2 + 1)-dimensional Maccari system are investigated. It is shown that two different types of breathers can be derived, and under appropriate parameter constraints, they can be reduced to some well known solutions, involving the homoclinic orbits, dark soliton or anti-dark soliton solution. For the dark and anti-dark solution, its interaction is similar to a resonance soliton. Furthermore, by using a limiting technique, we derive two kinds of rational solutions, one is the lump and the other one is the rogue wave. After constructing these solutions, we further discuss the interactions between the obtained solutions. It is interesting that we obtain a parallel breather and a intersectional breather, which seems very surprising. Finally, we also provide a new three-state interaction, which is composed by the dark-soliton, rogue wave and breather and has never been provided for the Maccari system.

The Influence of Multiplicative Noise and Fractional Derivative on the Solutions of the Stochastic Fractional Hirota–Maccari System

We address here the space-fractional stochastic Hirota–Maccari system (SFSHMs) derived by the multiplicative Brownian motion in the Stratonovich sense. To acquire innovative elliptic, trigonometric and rational stochastic fractional solutions, we employ the Jacobi elliptic functions method. The attained solutions are useful in describing certain fascinating physical phenomena due to the significance of the Hirota–Maccari system in optical fibers. We use MATLAB programm to draw our figures and exhibit several 3D graphs in order to demonstrate how the multiplicative Brownian motion and fractional derivative affect the exact solutions of the SFSHMs. We prove that the solutions of SFSHMs are stabilized by the multiplicative Brownian motion around zero.

Investigation into the Explicit Solutions of the Integrable (2+1)—Dimensional Maccari System via the Variational Approach

In this paper, the integrable (2+1)-dimensional Maccari system (MS), which can model many complex phenomena in hydrodynamics, plasma physics and nonlinear optics, is investigated by the variational approach (VA). This proposed approach that based on the variational theory and Ritz-like method can construct the explicit solutions via the stationary conditions only taking two steps. Finally, the dynamic behaviors of the solutions are exhibited by choosing the appropriate parameters through the 3-D and density plots. It can be seen that the proposed method is concise and straightforward, and can be adopted to study the travelling wave theory in physics.

Optimal system of Lie subalgebra for symmetry reductions, group invariant solutions and exact solutions to the coupled Hirota–Maccari system driving pulse propagation in optical fiber

In this paper, Lie symmetry analysis has been proposed by utilizing the Lie group of continuous point transformation for obtaining the new exact soliton solutions of the (2+1)-dimensional Hirota–Maccari system. Lie infinitesimals and possible geometric vector fields are obtained by applying the third-order prolongation on this system. Also, their commutative product and adjoint relations have been presented in Tables 1 and 2. By considering the resulting symmetries, one-dimensional optimal system of Lie subalgebra is obtained. Meanwhile, the Hirota–Maccari system is reduced to a system of ordinary differential equations with the help of optimal subalgebras. Furthermore, the simplest equation method has been used to obtain the abundant exact soliton solutions of the reduced system. At last, conservation laws of Hirota–Maccari system have been extracted by utilizing the generalized “new conservation theorem” invoked by Ibragimov.

Investigation of diverse genres exact soliton solutions to the nonlinear dynamical model via three mathematical methods

This work aims to study new exact soliton solutions to the nonlinear Maccari’s system using the new extended direct algebraic method (NEDAM), the unified method, and the extended Sinh-Gordon equation expansion method (ShGEEM). Diverse genres exact soliton solutions are secured in the form of bright, dark, singular and in their combined behavior as bright-dark, dark-singular and periodic wave solutions. The Maccari system is a nonlinear model that represents the dynamics of waves, confined to a small part of space, in different regions such as nonlinear optics, plasma physics, mathematical physics, fluid mechanics, and hydrodynamics, etc. The secured solutions contain vital applications in engineering and physics. These solutions define the wave performance of the governance models. All the obtained solutions verified the considered model in this study. By choosing different parametric values, we established 3D, 2D and contour profiles for some selected solutions to describe the dynamic physical behavior of the acquired solutions. The novelty of the obtained results is discussed with a detailed comparison with the already existing results. The proposed methods are powerful and can be applied alternatively to recover new soliton solutions of different types of partial differential equations.

Optical solitons and other solutions to the Hirota–Maccari system with conformable, M-truncated and beta derivatives

In this research paper, we scrutinize the novel traveling wave solutions and other solutions with conformable, M-truncated and beta fractional derivatives for the nonlinear fractional Hirota–Maccari system. In order to acquire the analytical solutions, the Riccati–Bernoulli sub-ODE technique is implemented. Presented method is the very powerful technique to get the novel exact soliton and other solutions for nonlinear partial equations in sense of both integer and fractional-order derivatives. Mathematical properties of different kinds of fractional derivatives are given in this paper. A comparative approach is presented between the solutions with the fractional derivatives. For the validity of the solutions, the constraints conditions are determined. To illustrate the physical meaning of the presented equation, the 2D and 3D graphs of the acquired solutions are successfully charted by selecting appropriate values of parameters.

Promulgation on various genres soliton of Maccari system in nonlinear optics

Promulgation on various genres soliton of Maccari system in nonlinear optics

New-fashioned solitons of coupled nonlinear Maccari systems describing the motion of solitary waves in fluid flow

The Schrodinger equation type nonlinear coupled Maccari system is a significant equation that flourished with the wide-ranging arena concerning fluid flow and the theory of deep-water waves, physics of plasma, nonlinear optics, etc. We exploit the enhanced tanh approach and the rational (G′/G)-expansion process to retrieve the soliton and dissimilar soliton solutions to the Maccari system in this study. The suggested systems of nonlinear equations turn into a differential equation of single variable through executing some operations of wave variable alteration. Thereupon, with the successful implementation of the advised techniques, a lot of exact soliton solutions are regained. The obtained solutions are depicted in 2D, 3D, and contour traces by assigning appropriate values of the allied unknown constants. These diverse graphical appearances assist the researchers to understand the underlying processes of intricate phenomena of the leading equations. The individual performances of the employed methods are praiseworthy which justify further application to unravel many other nonlinear evolution equations ascending in various branches of science and engineering.

Computational simulation and nonlinear vibration motions of isolated waves localized in small part of space

Nonlinear Schrödinger equations are examined to see if any new solution structures can be found for three different kinds of nonlinear equations. The generalized Korteweg–de Vries (gKdV) equation, the (2 + 1)–Ablowitz–Kaup–Newwell–Segur (AKNS) equation, and the Maccari system are all investigated. These models commonly characterize kinetic energy operators, which are affected by mass location and time dependence and the numerous representations of kinetic energy operators. Soliton wave solutions for the three new models are created by using the analytical Khater II approach. Figures depicting the numerical simulations of the resulting solutions may be seen all over the place. Mathematica 12 software is used to verify the approaches’ accuracy and re-incorporate all new results into the original models.

Some novel dynamical behaviours of localized solitary waves for the Hirota–Maccari system

Some novel dynamical behaviours of localized solitary waves for the Hirota–Maccari system

Weierstrass and Jacobi elliptic solutions with some new dromions to Maccari system

This paper studies Attilio Maccari (AM) system for some new solitary wave solutions. We used sub-ODE scheme to find kink shape, bell type, Weierstrass and Jacobi elliptic, hyperbolic, periodic and other solitary wave solutions under some constraint conditions. We will also present our results graphically in distinct dimensions.

Solitary wave solutions of (2+1)-dimensional Maccari system

In this article, three mathematical techniques have been operationalized to discover novel solitary wave solutions of (2+1)-dimensional Maccari system, which also known as soliton equation. This model equation is usually of applicative relevance in hydrodynamics, nonlinear optics and plasma physics. The [Formula: see text] function, the hyperbolic function and the [Formula: see text]-expansion techniques are used to obtain the novel exact solutions of the (2+1)-dimensional Maccari system (arising in nonlinear optics and in plasma physics). Many novel solutions such as periodic wave solutions by [Formula: see text] function method, singular, combined-singular and periodic solutions by hyperbolic function method, hyperbolic, rational and trigonometric solutions by [Formula: see text]-expansion method are obtained. The exact solutions are shown through 3D graphics which present the movement of the obtained solutions.

Rogue breathers and rogue lumps on a background of dark line solitons for the Maccari system

The Maccari system is a nonlinear dynamical model for the description of two-dimensional rogue waves occuring in diverse physical settings. For this two-dimensional nonlinear system, the time localized breathers and doubly localized lumps termed as rogue breathers and rogue lumps on a background of dark line solitons are investigated. The rogue breather behaves as a breather appearing and disappearing on a background of two dark line solitons, and is generated from the resonant collision between a breather and two dark line solitons. The limit case of a rogue breather is a rogue lump, which is truly localized in time and in the two-dimensional space. The Nth-order rogue lumps on a background of (N+1) dark line solitons are constructed by employing the Kadomtsev-Petviashvili hierarchy reduction method combined with the Hirota bilinear method. These higher-order lumps arise from a background of dark line solitons and then disappear into the background again after living for a very short period of time. The rogue lumps possess the key features of two-dimensional rogue waves occurring in a plethora of physical settings.