Dimensional Jimbo Miwa Equation Research Articles

Exact solutions of the (3+1)-dimensional Jimbo-Miwa equation via Wronskian solutions: Soliton, breather, and multiple lump solutions

In this paper, the Hirota bilinear method is applied to investigate the exact solutions of the (3+1)-dimensional Jimbo-Miwa (JM) equation, including solitons, breathers and lumps, which satisfy specific Wronskian conditions. Their dynamic behaviors and the effects of free parameters on the propagation direction and velocity are analyzed through three-dimensional images and the corresponding contour plots. Especially, based on the 2Mth-order Wronskian determinant solutions, the determinant expression of arbitrary Mth-order lump solutions is constructed by employing elementary transformation and long wave limit. The experimental results show that the interaction between multiple lumps is a completely elastic collision. These results may be helpful to understand the propagation processes of nonlinear waves in some nonlinear physical systems, such as fluid mechanics, nonlinear optics and so on.

(G'/G,1/G)-expansion method for analytical solutions of Jimbo-Miwa equation

The main goal of this study is obtaining analytical solutions for (3+1)-dimensional Jimbo-Miwa Equation which the second equation in the well-known KP hierarchy of integrable systems. For the (3+1DJM) equation, hyperbolic, trigonometric, complex trigonometric and rational traveling wave solutions have been constructed by applying the (G'/G,1/G)-expansion method. Then, real and imaginary graphics are presented by giving special values to the constants in the solutions obtained. These graphics are a special solution of the (3+1DJM) equation and represent a stationary wave of the equation. The (G'/G,1/G)-expansion method is an effective and powerful method for solving nonlinear evolution equations (NLEEs). Ready computer package program is used to obtain the solutions and graphics presented in this study.

Higher order rogue waves for the(3 + 1)-dimensional Jimbo–Miwa equation

Abstract Based on the Hirota bilinear form for the (3 + 1)-dimensional Jimbo–Miwa equation, we constructed the first-order, second-order, third-order and fourth-order rogue waves for this equation using the symbolic computation approach. Also some properties of the higher-order rogue waves and their interaction are explained by some figures via some special choices of the parameters.

Abundant lump-type solutions for the extended (3+1)-dimensional Jimbo–Miwa equation

In this work, we research the mixed lump–kink solutions and their dynamic properties of the extended (3+1)-dimensional Jimbo–Miwa equation. Mixed lump–kink solutions are triggered by the interaction between lump soliton and exponential function, the interaction between lump soliton and hyperbolic function, and the interaction between lump soliton and periodic wave. Their dynamics properties are shown by means of many graphics and corresponding density plots.

MEFM For Exact Solutions Of The (3+1) Dimensional KZK Equation and (3+1) Dimensional JM Equation

In this study, we have applied the modified exp -expansion function method (MEFM) to obtain the exact travelling wave solutions for the (3+1) dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation and (3+1)-dimensional Jimbo-Miwa (JM) equation. Dark soliton solution, dark-bright soliton solution, hyperbolic function solution and trigonometric function solution of the (3+1) dimensional KZK equation and (3+1)-dimensional JM equation have been found by using this method. After that, we have scratched the 2D and 3D graphs for all exact solutions obtained in this study by using Wolfram Mathematica 9. Thus, the graphical simulations openly show force of this method.

Breathers and multiple rogue waves solutions of the (3+1)-dimensional Jimbo–Miwa equation

In this paper, we construct the breathers of the (3+1)-dimensional Jimbo–Miwa (JM) equation by means of the Hirota bilinear method, then based on the Hirota bilinear method with a new ansatz form, the multiple rogue wave solutions are constructed. Here, we discuss the general breathers, first-order rogue waves, the second-order rogue waves and the third-order rogue waves. Then we draw the 3- and 2-dimensional plots to illustrate the dynamic characteristics of breathers and multiple rogue waves. These interesting results will help us better reveal (3+1)-dimensional JM equation evolution mechanism.

CHARACTERISTICS OF NEW TYPE ROGUE WAVES AND SOLITARY WAVES TO THE EXTENDED (3+1)-DIMENSIONAL JIMBO-MIWA EQUATION

<abstract> On the basis of the binary Bell polynomial scheme, the bilinear form of the extended (3+1)-dimensional Jimbo-Miwa (JM) equation, is constructed. Then, a class of new type rogue waves solutions to the extended (3+1)-dimensional JM equation, is found. It mainly includes the lump solutions, lumpoff solutions and instanton solutions. Their nonlinear evolutionary processes by 3D- and 2D-graphs, are shown. Finally, a direct method which is called the tanh-function method was used to get solitary waves of this considered model. These results can help us better understand interesting physical phenomena and mechanism. </abstract>

Rogue wave solutions and the bright and dark solitons of the (3+1)-dimensional Jimbo–Miwa equation

It is well known that most classical test functions to solve nonlinear partial differential equations can be constructed via single hidden layer neural network model by using Bilinear Neural Network Method (BNNM). In this paper, the neural network model of test function for the (3+1)-dimensional Jimbo–Miwa equation is extended to the “4-2-3” model. By giving some specific activation functions, new test function is constructed to obtain analytical solutions of the (3+1)-dimensional Jimbo–Miwa equation. Rogue wave solutions and the bright and dark solitons are obtained by giving some specific parameters. Via curve plots, three-dimensional plots, contour plots and density plots, dynamical characteristics of these waves are exhibited.

(1/G')-Expansion Method for Exact Solutions of (3+1)-Dimensional Jimbo-Miwa Equation

The purpose of this article is obtaining the exact solutions for (3+1)-dimensional Jimbo-Miwa Equation (3+1DJME). The (1/G')-expansion method which is an effective method in solving nonlinear evolution equations (NLEEs) is used. Then, 3D, contour and 2D graphics are presented by giving special values to the constants in the solutions obtained. These graphics are a special solution of the (3+1DJME) and represent a stationary wave of the equation. Ready computer package program is used to obtain the solutions and graphics presented in this study.

Investigation on breather waves and rogue waves in applied mechanics and physics

Nonlinear evolution equation is a research hot spot in the field of science and engineering. Recently, searching for breather and rogue wave solutions to nonlinear evolution equations has become a popular topic in nonlinear mathematical physics. In this paper, with the help of symbolic calculation, breather and rogue wave solutions to the two generalized (3+1)-dimensional Jimbo-Miwa equation are investigated, respectively. Those analytic solutions are presented, and the influences of the corresponding parameters on breathers and rogue waves are analyzed. Their dynamic properties are discussed based on graphical presentation. Amplitude of breathers and rogue waves are adjusted. Results of this study are helpful to the generation of breather and rogue waves in applied mechanics and physics.

Investigation of ion - acoustic wave dynamics in unmagnetized grain plasmas

In this work, we study an extended (3+1)-dimensional Jimbo-Miwa equation. We analyze the dynamics of three-dimensional nonlinear ion-acoustic waves in an unmagnetized plasma. We furnish the Lie symmetry technique to investigate the symmetry reductions of the examined model. The arbitrary functions involved in the Lie vectors are secured by using an optimization process, that leads to a reduction of the number of vectors in the commutator table. We next apply the singular manifold method (SMM) to the obtained reduced equations. This will result in a Schwarzian derivative whose solution will lead to new traveling wave solutions.

Forward scattering for non-linear wave propagation in (3 + 1)-dimensional Jimbo-Miwa equation using singular manifold and group transformation methods

Forward scattering of (3 + 1)-dimensional Jimbo-Miwa (JM) equation is investigated by singular manifold method (SMM). The detected Lax pair is reduced through three parameters group method into a compatible couple of ordinary differential equations (ODEs). Three forms of symmetry variables have been discussed leading to integrable system of ODEs. Some new closed form solutions have been obtained. The three parameters group method, which is rarely used in solving PDEs, has emphasized its efficiency in creating symmetry solutions of evolution equations.

New travelling wave and anti-kink wave solutions of space-time fractional (3+1)-Dimensional Jimbo–Miwa equation

In this work, the new analytical exact solution of a highly nonlinear fractional partial differential equation (FPDE) has been presented; specifically space-time fractional (3+1)-dimensional Jimbo–Miwa (JM) equation. As a consequence of the applied extended (G'/G)-expansion method, the new analytical exact solution of the governing equation has been acquired successfully. Moreover, the physical natures of the solutions have been analyzed here by means of numerical simulations.

Rational Solutions and Their Interaction Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation

In this paper, we gave a form of rational solution and their interaction solution to a nonlinear evolution equation. The rational solution contained lump solution, general lump solution, high-order lump solution, lump-type solution, etc. Their interaction solution contained the classical interaction solution, such as the lump-kink solution and the lump-soliton solution. As the example, by using the generalized bilinear method and symbolic computation Maple, we obtained abundant high-order lump-type solutions and their interaction solutions between lumps and other function solutions under certain constraints of the (3+1)-dimensional Jimbo-Miwa equation. Via three-dimensional plots, contour plots and density plots with the help of Maple, the physical characteristics and structures of these waves are described very well. These solutions have greatly enriched the exact solutions of the (3+1)-dimensional Jimbo-Miwa equation on the existing literature.

Novel interaction phenomena of the (3+1)-dimensional Jimbo–Miwa equation**Supported by the National Natural Science Foundation of China (11661060), Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-20-A06) and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (2018LH01013).

In this paper, we give the general interaction solution to the (3+1)-dimensional Jimbo–Miwa equation. The general interaction solution contains the classical interaction solution. As an example, by using the generalized bilinear method and symbolic computation by using Maple software, novel interaction solutions under certain constraints of the (3+1)-dimensional Jimbo–Miwa equation are obtained. Via three-dimensional plots, contour plots and density plots with the help of Maple, the physical characteristics and structures of these waves are described very well. These solutions greatly enrich the exact solutions to the (3+1)-dimensional Jimbo–Miwa equation found in the existing literature.

Lump-type and interaction solutions to an extended (3 + 1)-dimensional Jimbo–Miwa equation

By using the Hirota bilinear method, we construct new lump-type solutions to an extended [Formula: see text]-dimensional Jimbo–Miwa equation, which describes certain [Formula: see text]-dimensional wave phenomena in physics. The presented solutions contain 10 arbitrary parameters and only need to satisfy four restrictive conditions to be analytic, thereby enriching the existing lump-type solutions. Moreover, we compute their interaction solutions with double exponential function waves, which include rogue wave solutions. Dynamical features of the obtained solutions are graphically exhibited.

High-order lumps, high-order breathers and hybrid solutions for an extended (3 + 1)-dimensional Jimbo–Miwa equation in fluid dynamics

Under investigation in this letter is an extended (3 + 1)-dimensional Jimbo–Miwa (eJM) equation, which can be used to describe many nonlinear phenomena in mathematical physics. With the aid of Hirota bilinear method and long-wave limit method, M-order lumps which describe multiple collisions of lumps are derived. The propagation orbit, velocity and extremum of the 1-order lump solutions on (x, y) plane are investigated in detail. Resorting to the extended homoclinic test technique, we obtain the breather–kink solutions, rational breather solutions and rogue wave solutions for the eJM equation. Meanwhile, through analysis and calculation, the amplitude and period of breather–kink solutions increase with p increasing and the extremum of rational breather solution and rogue waves are also derived. T-order breathers are obtained by means of choosing appropriate complex conjugate parameters on N-soliton solutions. Periods of the 1-order breather solutions on the (x, y) plane are determined by $$k_{12}$$ and $$k_{12}p_{11}+k_{11}p_{12}$$, while locations are determined by $$k_{11}$$ and $$k_{11}p_{11}-k_{12}p_{12}$$. Furthermore, hybrid solutions composed of the kink solitons, breathers and lumps for the eJM equation are worked out. Some figures are given to display the dynamical characteristics of these solutions.

Semi-rational and periodic wave solutions for the (3+1)-dimensional Jimbo–Miwa equation

In this paper, higher-order periodic wave solution in determinant form is investigated via the Kadomtsev–Petviashvili hierarchy reduction for the $$(3+1)$$-dimensional Jimbo–Miwa equation. We obtain the breather and periodic wave via such solution. The breather is periodic in space and localized in time, while amplitude of the periodic wave changes with the value of y. Besides, interaction between the breather and periodic wave shows a different characteristic in which amplitude of the periodic wave changes after the interaction. By taking the long wave limit to the periodic wave solution, we obtain the semi-rational solution, and then derive the rogue wave and kink-like soliton. Furthermore, we conclude that (1) rogue wave can be obtained with the limit on the breather; (2) kink-like soliton can be obtained with the limit on the periodic wave. Via the semi-rational solution, interactions between the rogue wave and breather, between the kink-like soliton and breather, are illustrated.

A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation

This paper aims to study a generalized first extended (3+1)- dimensional Jimbo-Miwa equation. Symmetry reductions on this equation are performed several times and it is reduced to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is found in terms of the incomplete elliptic integral function. Also exact solutions are constructed using the \begin{document}$ ({G'}/{G})- $\end{document} expansion method. Thereafter the conservation laws of the underlying equation are computed by invoking the conservation theorem due to Ibragimov. The conservation laws obtained contain an energy conservation law and three momentum conservation laws.

On integrability of the extended (3+1)‐dimensional Jimbo‐Miwa equation

By applying the binary bell polynomial scheme, the bilinear form, Bäcklund Transformations, and lax pairs of an extended (3+1)‐dimensional Jimbo‐miwa (JM) equation are constructed. Next, periodic wave‐type solutions can also be obtained to the extended (3+1)‐dimensional JM equation through the three‐wave method with the help of maple. Finally, a test function of the sech‐function method is utilized to get solitary waves of this study problem. These new results can help us better understand interesting physical phenomena and mechanism.