Dimensional Jimbo Miwa Equation Research Articles

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.

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.

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.

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.

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.

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

Classification of All Single Traveling Wave Solutions of ( 3 + 1 )-Dimensional Jimbo-Miwa Equation with Space-Time Fractional Derivative

In this paper, the complete discrimination system method is used to construct the single traveling wave solutions for the ( 3 + 1 )-dimensional Jimbo-Miwa equations with space-time fractional derivative. As a result, we get the exact traveling wave solutions of the ( 3 + 1 )-dimensional Jimbo-Miwa equation with space-time fractional derivative, which include rational function solutions, Jacobian elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. Some graphical representations of the solutions are also provided. Finally, the obtained solution is compared with the existing literature.

The ∂̄-dressing method for the (2+1)-dimensional Jimbo-Miwa equation

The (2+1)-dimensional Jimbo-Miwa equation is analyzed by means of the ∂ ¯ \bar {{\partial }} -dressing method. By means of the characteristic function and Green’s function of the Lax representation, the problem has been transformed into a new ∂ ¯ \bar {{\partial }} problem. A solution is constructed based on solving the ∂ ¯ \bar {{\partial }} problem with the help of Cauchy-Green formula and choosing the proper spectral transformation. Furthermore, we can obtain the solution formally of the Jimbo-Miwa equation when the time evolution of the spectral data is determined.

On multiple soliton solutions of the extended (3+1)-dimensional Jimbo–Miwa equations

In this paper, we deal with a set of equations, namely the extended (3 + 1)-dimensional Jimbo–Miwa (JM) equations. We focus on the calculations of soliton solutions to the aforementioned equations using the multiple exponential function method (MEF). As a result, we derive up the one to three-wave soliton solution to the both equations, while presenting the profiles obtained through the constant values related to those solutions. It is noteworthily to point out that, all the calculations in this paper are made with software package Maple. Also, the profiles of the waves depend on the arbitrary values of autocephalous parameters.

Investigation of Nonlinear Wave Solutions for Fusion and Fission Phenomenas

In this study, wave solutions of the (3+1) dimensional Jimbo-Miwa equation and two different phenomena of the solution, fusion and fission, are obtained using the modified exponential functionmethod. In order to get more possible solutions, two different cases are investigated due to the nature of the modified exponential function method. When the resulting solutions are analyzed, trigonometric, hyperbolic and rational functions are obtained. It was checked whether the solution functions found by the Wolfram Mathematica software provided the (3+1) dimensional potential Jimbo-Miwa equation. Two and three dimensional graphs, contour and density graphs of the solution function were get by determining the appropriate parameters.

Wronskian and Grammian conditions, and Pfaffianization of an extended (3+1)-dimensional Jimbo–Miwa equation

In this paper, a Wronskian condition and a Grammian condition are proposed for an extended (3+1)-dimensional Jimbo–Miwa equation, which guarantee that the N-soliton solutions of the equation can be expressed in Wronskian and Grammian determinants. Moreover, some examples of Wronskian and Grammian solutions are explicitly computed, and the corresponding soliton dynamics are graphically illustrated via Mathematica by choosing suitable parameters. In addition, the Pfaffianization of the extended (3+1)-dimensional Jimbo–Miwa equation is realized to generate a new coupled (3+1)-dimensional system whose solutions are further expressed as Wronskian-type and Grammian-type Pfaffians.

An extended (3+1)-dimensional Jimbo–Miwa equation: Symmetry reductions, invariant solutions and dynamics of different solitary waves

The Lie symmetry method is used to obtain a variety of closed-form wave solutions for the extended (3[Formula: see text]+[Formula: see text]1)-dimensional Jimbo–Miwa (JM) Equation, which describes certain interesting higher-dimensional waves in ocean studies, marine engineering, and other fields. By applying the Lie symmetry technique, we explicitly investigate all the possible vector fields, commutation relations of the considered vectors, and various symmetry reductions of the equation. Based on three stages of Lie symmetry reductions, the JM equation is reduced to several nonlinear ordinary differential equations (NLODEs). Consequently, abundant closed-form wave solutions are achieved, including arbitrary functional parameters. Evolutionary dynamics of some analytic wave solutions are demonstrated through three-dimensional plots based on numerical simulation. Consequently, singular soliton, kink waves, periodic oscillating wave profiles, combined singular soliton profiles, curved-shaped multiple solitons, and periodic multiple solitons with parabolic wave profiles are demonstrated by taking advantage of symbolic computation work. The obtained analytical wave solutions, which include arbitrary independent functions and other constants of the governing equation, could be used to enrich the advanced dynamical behaviors of solitary wave solutions. Furthermore, the study of conservation laws is investigated via the Ibragimov technique for Lie point symmetries.

Some new soliton solutions and dynamical behaviors of (3+1)-dimensional Jimbo-Miwa equation

Some new solitary solutions of (3+1)-dimensional Jimbo-Miwa equation such as breather solutions, double breather solutions and mixed solutions of different forms are studied via applying Hirota's bilinear form and constructing various trial functions. Besides, the degradation phenomenon of breather solution to lump-type solution is investigated, and dynamical behaviors of the interaction between solitary solutions and lump-type solutions varying with time t is discussed. Finally, the interactive result between N-soliton solution (N takes any positive integer) and lump-type solution is studied by reducing dimension, and the detailed proof process is presented.

Some New Traveling Wave Solutions of Nonlinear Fluid Models via the MSE Method

In this study, some new exact wave solutions of nonlinear partial differential equations are investigated by the modified simple equation method. This method is applied to the $(2+1)$-dimensional Calogero-Bogoyavlenskii-Schiff equation and the $(3+1)$-dimensional Jimbo-Miwa equation. Our applications reveal how to use the proposed method to solve nonlinear partial differential equations with the balance number equal to two. Consequently, some new exact traveling wave solutions of these equations are achieved, and types of waves are determined. To verify our results and draw the graphs of the solutions, we use the Mathematica package program.

Generalised exponential rational function method for obtaining numerous exact soliton solutions to a ($$3+1$$)-dimensional Jimbo–Miwa equation

Generalised exponential rational function method for obtaining numerous exact soliton solutions to a ($$3+1$$)-dimensional Jimbo–Miwa equation

Soliton molecules and some interaction solutions for the (3+1)-dimensional Jimbo-Miwa equation

Soliton solutions, lump solutions, breather solutions and soliton molecules are hot topics in recent years. And soliton molecules have been successfully discovered in optical experiments. In this paper, we firstly use the Hirota bilinear method to obtain the multiple solitary wave solutions of the (3+1)-dimensional Jimbo-Miwa equation. Secondly, on the basis of soliton solutions, the soliton molecule of the equation is studied by means of the velocity resonance condition, and the dynamic diagram of the specific case is given. Finally, we study the hybrid solution of the equation involving soliton, lump, breather, and soliton molecule.

On the dynamics of localized excitation wave solutions to an extended (3+1)-dimensional Jimbo–Miwa equation

Three types of localized excitation structures, including multi-dromion, multi-lump and periodic solutions, for the extended (3+1)-dimensional Jimbo–Miwa equation are constructed based on Hirota bilinear and variable separation methods. By adding constraint conditions to make the velocities equal or not, a bound state solution can be obtained, which demonstrates chase collision and interesting interaction phenomena. Especially, their dynamic characteristics are analyzed in detail and vividly displayed.