Jimbo Miwa Equation Research Articles

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.

Solitons, travelling waves, invariance, conservation laws and ‘approximate’ conservation of the extended Jimbo-Miwa equation

Abstract We will make a detailed analysis of a class of the Jimbo-Miwa equation, viz., exact or invariant solutions that arise from the symmetries generated by it and the conservation laws of the equation. The equation, it turns out, is rich in its symmetry structure and produces a large class of conservation laws and what we term as ‘approximate’ conservation laws.

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.

Nonsingular complexiton solutions and resonant waves to an extended Jimbo–Miwa equation

The goal of this letter is to consider an extended Jimbo–Miwa equation which exists nonsingular complexiton solutions in (3+1)-dimensions. A few types of two-wave and complexiton solutions are developed through symbolic computations with Maple, including resonant two-wave solutions and nonsingular complexiton solutions. Nonsingular complexitons to the generalized Jimbo–Miwa equation should be new, though there exist plenty of studies in the literature. A few specific cases are derived to illustrate the remarkable richness of explicit solutions for the considered equation.

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.

Multiwave interaction solutions for the (3+1)-dimensional extended Jimbo–Miwa equation

In this paper, for the (3+1)-dimensional extended Jimbo–Miwa equation, by the direct algebraic method, together with the inheritance solving strategy, we construct its interaction solutions among solitons, rational waves, and periodic waves. Meanwhile, we construct its interaction solutions among solitons, breathers, and lumps of any higher orders by an [Formula: see text]-soliton decomposition algorithm, together with the parameters conjugated assignment and long-wave limit techniques. The highlight of the paper is that by applying new algorithms and new techniques, we obtained different types of new multiwave interaction solutions for the (3+1)-dimensional extended Jimbo–Miwa equation.

Localized characteristics of lump and interaction solutions to two extended Jimbo–Miwa equations* *Project supported by the Fundamental Research Funds for the Central Universities of China (Grant No. 2018RC031), the National Natural Science Foundation of China (Grant No. 71971015), the Program of the Co-Construction with Beijing Municipal Commission of Education of China (Grant Nos. B19H100010 and B18H100040), and the Open Fund

We focus on the localized characteristics of lump and interaction solutions to two extended Jimbo–Miwa equations. Based on the Hirota bilinear method and the test function method, we construct the exact solutions to the extended equations including lump solutions, lump–kink solutions, and two other types of interaction solutions, by solving the under-determined nonlinear system of algebraic equations for associated parameters. Finally, analysis and graphical simulation are presented to show the dynamical characteristics of our solutions and the interaction behaviors are revealed.

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.

Exact Solutions for Time-Fractional Ramani and Jimbo—Miwa Equations by Direct Algebraic Method

In this study, the analytical solutions of some nonlinear time-fractional partial differential equations are investigated by the direct algebraic method. The nonlinear fractional partial differential equation (NLfPDE) which is based on the fractional derivative (fd) in the sense of modified Riemann-Liouville derivative is transformed to the nonlinear non-fractional ordinary differential equation. The hyperbolic and rational functions which are contained solutions are obtained for the sixth-order time-fractional Ramani equation and time-fractional Jimbo—Miwa equation (JME) with the help of this technique. In addition, this method can be applied to higher order and higher dimensional NLfPDEs. Finally, three dimensional simulations of some solutions are given.

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.

Periodic oscillating solitons and homoclinic breather-wave solution for the (3+1)-dimensional Jimbo-Miwa equation

With the aid of symbolic computation, some new types of breathing wave solutions to a (3+1)-D Jimbo-Miwa equation are obtained by the extended homoclinic test method. Its homoclinic breather-wave solution, periodic oscillating soliton and doubly-soliton solution are investigated.

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.