Jimbo Miwa Equation Research Articles

The bifurcation, chaotic behavior and exact solutions of the fractional stochastic Jimbo–Miwa equations

The Jimbo–Miwa equation is the second equation in the KP hierarchy of integrable systems. In this paper, this equation is extended and introduced with the stochastic process and fractional derivatives. Firstly, the phase portrait of the Hamiltonian system generated by it is studied to understand its bifurcation behavior. Additionally, non-periodic and periodic perturbation terms are added to this system. Different values are assigned to the parameters in the perturbation terms to analyze its sensitivity and the resulting chaos is obtained. Finally, through integration techniques, the expression of the solution of this equation is obtained. These solutions are related to rational functions, trigonometric functions, exponential functions and Jacobi elliptic functions. To observe the form of the solutions more intuitively, 3D and 2D numerical simulations are conducted on the solutions and the solution images of the stochastic fractional differential equation are given by Matlab software. Compared with the existing literature, the research on the stochastic fractional equation of this equation is relatively rare and the analysis of the phase portrait is even scarcer. Our solution method is quite different from that in the previous literature. Therefore, this paper is novel. The conclusion of this paper will be of great help for the practical application of this equation.

New Multilinear Variable Separation Solutions of the (3 + 1)-Dimensional Burgers Hierarchy

The multilinear variable separation (MLVS) approach has been proven to be very useful in solving (2 + 1)-dimensional integrable systems. Taking the (3 + 1)-dimensional Burgers hierarchy as an example, we extend the MLVS approach to a whole family of (3 + 1)-dimensional Burgers hierarchy. New exact solutions and universal formulas are obtained, which lead to abundant (3 + 1)-dimensional coherent structures. In particular, two ring-type soliton molecules and their interactions are shown in detail. We also generalize the MLVS results of the (3 + 1)-dimensional Jimbo–Miwa (JM) equation and modified JM equation.

Dynamical analysis and extraction of solitonic structures of a novel model in shallow water waves

In this study, we utilized a novel auto-Bäcklund transformation and the extended transformed rational function approach to analyze the extended reduced Jimbo–Miwa equation, a prominent equation within the KP hierarchy. The homogenous balance technique was employed to derive the auto-Bäcklund transformation of the equation, leading to the extraction of new exact solutions exhibiting solitary patterns. Additionally, we applied the extended transformed rational function method, which relies on the Hirota bilinear form of the governing equation, to generate complexiton solutions. Furthermore, we included 3D graphics visualizing the obtained solutions.

Travelling Wave Solutions for Some Time-Fractional Nonlinear Differential Equations

This study employs the powerful generalized Kudryashov method to address the challenges posed by fractional differential equations in mathematical physics. The main objective is to obtain new exact solutions for three important equations: the (3+1)-dimensional time fractional Jimbo-Miwa equation, the (3+1)-dimensional time fractional modified KdV-Zakharov-Kuznetsov equation, and the (2+1)-dimensional time fractional Drinfeld-Sokolov-Satsuma-Hirota equation. The generalized Kudryashov method is highly versatile and effective in addressing nonlinear problems, making it a pivotal component in our research. Its adaptability makes it useful in diverse scientific disciplines. The method simplifies complex equations, improving our analytical capabilities and deepening our understanding of system dynamics. Additionally, we define fractional derivatives using the conformable fractional derivative framework, providing a strong foundation for our mathematical investigations. This paper examines the effectiveness of the generalized Kudryashov method in solving complex challenges presented by fractional differential equations and aims to provide guidance for future studies.

Soliton molecules, Y-type soliton and complex multiple soliton solutions to the extended (3+1)-dimensional Jimbo-Miwa equation

Abstract The central purpose of this paper is extracting some novel and interesting soliton solutions of the extended (3+1)-dimensional Jimbo-Miwa equation(JME) which acts as an extension of the classic (3+1)-dimensional JME for the plasma and optics. First, we study the N-soltion solutions that is developed by the Hirota bilinear method (HBM). Then, the soliton molecules and Y-type soliton solutions are constructed via imposing the novel resonance conditions to the N-soltion solutions. In addition, we also explore the complex multiple soliton solutions via the HBM. The dynamic properties of the N-soltion, soliton molecules, Y-type soliton as well as the complex multiple soliton solutions are presented graphically. The developed soliton solutions of this research are all new and can enable us apprehend the nonlinear dynamic behaviors of the extended (3+1)-dimensional JME better.

SEMI-DOMAIN SOLUTIONS TO THE FRACTAL (3+1)-DIMENSIONAL JIMBO–MIWA EQUATION

In this work, the variational direct method (VDM) based on the variational theory and Ritz-like method, combined with the fractal two-scale transformation, is employed to construct the semi-domain solutions of the (3+1)-dimensional fractal Jimbo–Miwa (JM) equation, which is derived by He’s fractal derivative. The applicability and effectiveness of the acquired solutions on the Cantor sets are presented through the numerical results in the form of 3-dimensional graphs. In addition, as a comparison, the exact solutions of the classic (3+1)-dimensional JM equation for [Formula: see text] are also illustrated. The obtained results in this work are expected to open new perspectives for the traveling wave theory.

Novel soliton molecule solutions for the second extend (3+1)-dimensional Jimbo-Miwa equation in fluid mechanics

The main aim of this paper is to investigate the different types of soliton molecule solutions of the second extend (3+1)-dimensional Jimbo-Miwa equation in a fluid. Four different localized waves: line solitons, breather waves, lump solutions and resonance Y-type solutions are obtained by the Hirota bilinear method directly. Furthermore, the molecule solutions consisting of only line waves, breathers or lump waves are generated by combining velocity resonance condition and long wave limit method. Also, the molecule solutions such as line-breather molecule, lump-line molecule, lump-breather molecule, etc. consisting of different waves are derived. Meanwhile, higher-order molecule solutions composed of only line waves are acquired.

A deterministic approach to investigate nonlinear evolution equations for large balance numbers

When the balance number is greater than one, the modified simple equation (MSE) method typically fails to yield analytical wave solutions for nonlinear evolution equations (NLEEs) that appear in engineering and mathematical physics. We have addressed this shortcoming in this article and established a technique to implement the MSE approach to investigate NLEEs for balancing number two. Two NLEEs, namely, the regularized long wave and the Jimbo-Miwa equations, have been investigated in order to affirm the approach. Through this method, we found further generic wave solutions related to physical parameters, and when the parameters receive particular values, solitons emerge from the exact solutions. Graphs are used to investigate the solitary wave features of the attained solution functions, which illustrate the usefulness, validity, and compatibility of the scheme.

On soliton solutions, periodic wave solutions, asymptotic analysis and interaction phenomena of the (3+1)-dimensional JM equation

In this paper, the M-lump solutions, the periodic type, and cross-kink wave solutions are acquired. Here, the Hirota bilinear operator is employed. By utilizing the symbolic computation and employing the utilized method, the (3+1)-dimensional Jimbo–Miwa (JM) equation is investigated. Based on the Hirota bilinear form, the soliton solution and periodic wave solution to the mentioned equation, respectively, are obtained. We gained plenty of multiple collisions of lumps. Next, the periodic wave and cross-kink wave have greatly enriched the existing literature on the JM equation. Through the three-dimensional designs, contour design, density design, and two-dimensional design by using Maple, the physical features of these soliton solutions are explained all right. The forms of the attained solutions are one-lump, two-lumps, and three-lumps wave solutions. Then, a class of rogue waves-type solutions to the (3+1)-dimensional JM equation within the frame of the bilinear equation is found. These results can help us better understand interesting physical phenomena and mechanisms.

Localized waves and interaction solutions to an integrable variable coefficients Jimbo-Miwa equation

In this paper, the reduced variable coefficients Jimbo-Miwa (vcJM) equation is studied. Firstly, the integrability of the reduced vcJM equation is verified by Painlevé analysis. Based on the Hirota bilinear method and the long wave limit method, the N-soliton solutions, rational and semirational solutions of the vcJM equation are obtained. By choosing different parameters and coefficient functions, some of different kinds of local waves, including of solition, breather wave and lumps, of the equation are obtained. Furthermore, the interaction solutions between different local waves are obtained. The dynamical behavior of the interaction between different local waves is studied by modifying the time parameters and the process is displayed by figures.

Study on the Nonlinear Dynamics of the (3+1)-Dimensional Jimbo-Miwa Equation in Plasma Physics

The Jimbo-Miwa equation (JME) that describes certain interesting (3+1)-dimensional waves in plasma physics is studied in this work. The Hirota bilinear equation is developed via the Cole-Hopf transform. Then, the symbolic computation, together with the ansatz function schemes, are utilized to seek exact solutions. Some new solutions, such as the multi-wave complexiton solution (MWCS), multi-wave solution (MWS) and periodic lump solution (PLS), are successfully constructed. Additionally, different types of travelling wave solutions (TWS), including the dark, bright-dark and singular periodic wave solutions, are disclosed by employing the sub-equation method. Finally, the physical characteristics and interaction behaviors of the extracted solutions are depicted graphically by assigning appropriate parameters. The obtained outcomes in this paper are more general and newer. Additionally, they reveal that the used methods are concise, direct, and can be employed to study other partial differential equations (PDEs) in physics.

Exact Solutions of an Extended Jimbo-Miwa Equation by Three Distinct Methods

In this article, we focus on exact traveling wave solutions to an extended Jimbo-Miwa equation, which is an extension of the Jimbo-Miwa equation. First, an improved G ′ / G -expansion method, extended G ′ / G -expansion method, and improved two variable ( φ ′ / φ , 1 / φ ) expansion method are introduced. Second, with these introduced methods, many new exact traveling wave solutions of EJM equation are constructed, including hyperbolic function solutions, trigonometric function solutions, and rational function solutions which contain many different parameters. Finally, we depict the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. To the best of our knowledge, the received results have not been reported in other studies on the new extended JM equations. We hope that our results can help enrich the study of this new equation.

The Solitary Solutions for the Stochastic Jimbo–Miwa Equation Perturbed by White Noise

We study the (3+1)-dimensional stochastic Jimbo–Miwa (SJM) equation induced by multiplicative white noise in the Itô sense. We employ the Riccati equation mapping and He’s semi-inverse techniques to provide trigonometric, hyperbolic, and rational function solutions of SJME. Due to the applications of the Jimbo–Miwa equation in ocean studies and other disciplines, the acquired solutions may explain numerous fascinating physical phenomena. Using a variety of 2D and 3D diagrams, we illustrate how white noise influences the analytical solutions of SJM equation. We deduce that the noise destroys the symmetry of the solutions of SJM equation and stabilizes them at zero.

Mixed Lump-Stripe Soliton Solutions to a New Extended Jimbo-Miwa Equation

In this paper, the localized properties of lump and interaction solutions to a new extended Jimbo-Miwa (EJM) equation are studied. Based on the Hirota bilinear method and the test function method, the exact solutions of the EJM equation are discussed; the lump soliton solution, lump-kink soliton solution, and periodic lump solution are obtained. Furthermore, the dynamic properties of the obtained solutions are also discussed by graphical simulation. As far as we know, the obtained results have not been reported.

On soliton solutions, periodic wave solutions and asymptotic analysis to the nonlinear evolution equations in (2+1) and (3+1) dimensions

In this paper, the (2+1)-dimensional generalized fifth-order KdV equation and the extended (3+1)-dimensional Jimbo-Miwa equation were transformed into the Hirota bilinear forms with Hirota direct method. In this process, the Hirota bilinear operator played a significant role. Based on the Hirota bilinear forms, the single soliton solutions and the single periodic wave solutions of these two types of equations were obtained respectively. Meanwhile, the figures of the single soliton solutions and the single periodic wave solutions were plotted. Furthermore, the results shed light on that when the amplitude of water wave approaches 0, the single periodic wave solutions tend to the single soliton solutions. The conclusion has been generalized from (2+1)-dimensional equations to (3+1)-dimensional equations.

Interaction solutions for the second extended (3+1)-dimensional Jimbo–Miwa equation

Based on the Hirota bilinear method, the second extended (3+1)-dimensional Jimbo–Miwa equation is established. By Maple symbolic calculation, lump and lump-kink soliton solutions are obtained. The interaction solutions between the lump and multi-kink soliton, and the interaction between the lump and triangular periodic soliton are derived by combining a multi-exponential function or trigonometric sine and cosine functions with quadratic functions. Furthermore, periodic-lump wave solution is derived via the ansatz including hyperbolic and trigonometric functions. Finally, 3D plots, 2D curves, density plots, and contour plots with particular choices of the suitable parameters are depicted to illustrate the dynamical features of these solutions.

Solitary Solution of Jimbo–Miwa Equation by the Modified Extended and Multiple Exp-Function Methods

Solitary Solution of Jimbo–Miwa Equation by the Modified Extended and Multiple Exp-Function Methods

M-lump waves and their interactions with multi-soliton solutions for the (3 + 1)-dimensional Jimbo–Miwa equation

Abstract In this work, the dynamical behaviors of the Jimbo–Miwa equation that describes certain interesting (3 + 1)-dimensional waves in physics but does not pass any of the conventional integrability tests are studied. One-, two-, and three-M-lump waves are constructed successfully. Interactions between one-M-lump and one-soliton wave, between one-M-lump and two-soliton wave as well as between two-M-lump and one-soliton solution are reported. Also, complex multi-soliton, solutions are offered. The simplified Hirota’s method and a long-wave method are used to construct these types of solutions. The velocity of a one-M-lump wave is studied. Straight Lines of travel for M-lump waves are also reported. To our knowledge, all gained solutions in this research paper are novel and not reported beforehand. Moreover, the gained solutions are presented graphically in three dimensions to better understand the physical phenomena of the suggested equation.

EXACT TRAVELING WAVE SOLUTIONS TO THE LOCAL FRACTIONAL (3 + 1)-DIMENSIONAL JIMBO–MIWA EQUATION ON CANTOR SETS

In this work, we derive a new fractional [Formula: see text]-dimensional Jimbo–Miwa equation via the local fractional derivative. With the help of the traveling wave transform of the non-differentiable type, the local fractional [Formula: see text]-dimensional Jimbo–Miwa equation is converted into a nonlinear local fractional ordinary differential equation. Then, a new method named as Mittag-Leffler function-based method is used to construct the exact traveling wave solutions for the first time. Four families (eight sets) of the traveling wave solutions are obtained and the behaviors of the solutions on Cantor sets are presented via the 3D plot. The results show that the proposed method is a powerful tool to study the traveling wave theory of the local fractional equations.

General high-order rational solutions and their dynamics in the (3+1)-dimensional Jimbo–Miwa equation

General high-order rational solutions and their dynamics in the (3+1)-dimensional Jimbo–Miwa equation