Jimbo Miwa Equation Research Articles

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.

On solutions of the combined KdV-nKdV equation

The aim of this work is to deal with a new integrable nonlinear equation of wave propagation, the combined of the Korteweg-de vries equation and the negative order Korteweg-de vries equation (combined KdV-nKdV) equation, which was more recently proposed by Wazwaz. Upon using wave reduction variable, it turns out that the reduced combined KdV-nKdV equation is alike the reduced (3+1)-dimensional Jimbo Miwa (JM) equation, the reduced (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (PYTSF) equation and the reduced (3 + 1)¬dimensional generalized shallow water (GSW) equation in the trav¬elling wave. In fact, the four transformed equations belong to the same class of ordinary differential equation. With the benefit of a well known general solutions for the reduced equation, we show that sub¬jects to some scaling and change of parameters, a variety of families of solutions are constructed for the combined KdV-nKdV equation which can be expressed in terms of rational functions, exponential functions and periodic solutions of trigonometric functions and hyperbolic func¬tions. In addition to that the equation admits solitary waves, and double periodic waves in terms of special functions such as Jacobian elliptic functions and Weierstrass elliptic functions.

Multi-exponential wave solutions to two extended Jimbo–Miwa equations and the resonance behavior

Resonance phenomena occur widely in fluid, physics and other fields, e.g., they are related with the optical elements, the well-balanced scheme for shallow water with discontinuous topography, and some phenomena in chaotic dynamics and fluid dynamics. Application of the principle of linear superposition to the Hirota bilinear equation gives rise to a sufficient and necessary condition for the existence of multi-exponential wave solutions. We study the resonance behavior based on the construction of multi-exponential wave solution to two extended (3 + 1)- dimensional Jimbo–Miwa equations. The resonance characteristics are analyzed and simulated for some resonant two-wave and three-wave solutions.

Further investigations to extract abundant new exact traveling wave solutions of some NLEEs

In this study, we implement the generalized (G′/G)-expansion method established by Wang et al. to examine wave solutions to some nonlinear evolution equations. The method, known as the double (G′/G, 1/G)-expansion method is used to establish abundant new and further general exact wave solutions to the (3 + 1)-dimensional Jimbo–Miwa equation, the (3 + 1)-dimensional Kadomtsev–Petviashvili equation and symmetric regularized long wave equation. The solutions are extracted in terms of hyperbolic function, trigonometric function and rational function. The solitary wave solutions are constructed from the obtained traveling wave solutions if the parameters received some definite values. Graphs of the solutions are also depicted to describe the phenomena apparently and the shapes of the obtained solutions are singular periodic, anti-kink, singular soliton, singular anti-bell shape, compaction etc. This method is straightforward, compact and reliable and gives huge new closed form traveling wave solutions of nonlinear evolution equations in ocean engineering.

Exact traveling wave solutions to higher order nonlinear equations

The present paper applies the new generalized (G′/G)-expansion method on three non-linear equations including the fifth-order Korteweg-de Vries equation, (3+1)-dimensional Modified KdV-Zakharov-Kuznetsov equation, and (3+1)-dimensional Jimbo-Miwa equations and obtains some new exact traveling wave solutions. The solutions are checked by putting back into the physical model and are observed to be very advantageous over several existing methods. The newly method could also be utilized to find exact traveling wave solutions for the other nonlinear evolution equations in the field of applied nonlinear sciences.

Lump and interaction solutions of nonlinear partial differential equations

In this paper, lump solutions of nonlinear partial differential equations, the generalized (2[Formula: see text]+[Formula: see text]1)-dimensional KP equation and the Jimbo–Miwa equation, are studied by using the Hirota bilinear method and carrying out symbolic computations in Maple. Moreover, the interaction solutions, i.e. collisions between lump waves and kink waves, are investigated. A group of graphs are plotted to illustrate the dynamics of the obtained results.

Explicit Lump Solitary Wave of Certain Interesting (3+1)-Dimensional Waves in Physics via Some Recent Traveling Wave Methods.

This study investigates the solitary wave solutions of the nonlinear fractional Jimbo–Miwa (JM) equation by using the conformable fractional derivative and some other distinct analytical techniques. The JM equation describes the certain interesting (3+1)-dimensional waves in physics. Moreover, it is considered as a second equation of the famous Painlev’e hierarchy of integrable systems. The fractional conformable derivatives properties were employed to convert it into an ordinary differential equation with an integer order to obtain many novel exact solutions of this model. The conformable fractional derivative is equivalent to the ordinary derivative for the functions that has continuous derivatives up to some desired order over some domain (smooth functions). The obtained solutions for each technique were characterized and compared to illustrate the similarities and differences between them. Profound solutions were concluded to be powerful, easy and effective on the nonlinear partial differential equation.

Abound rogue wave type solutions to the extended (3+1)-dimensional Jimbo–Miwa equation

Under investigation in this article is an extended (3+1)-dimensional Jimbo–Miwa equation (JM), which can be used to describe many nonlinear phenomena in fluid dynamics. By using the Hirota bilinear form of the extended (3+1)-dimensional JM equation, thirty classes of rogue wave type solutions are found with the help of symbolic computations. The rogue wave type solutions contain two important parameters a and b. When a and b get different values, we can present obvious rogue wave type solutions. For example, (i) taking a=b=0, then we have algebraic solitary waves (lump); (ii) if one of a and b is fixed at 0 and the other is nonzero, then we have an interaction solution between an algebraically decayed soliton and exponentially decayed soliton (lumpoff); (iii) let a≠0 and b≠0, then we have an interaction solution between lump and exponentially localized twin soliton wave (instant or rogue wave). The new rogue wave type solutions help us to know different physical worlds.

New interaction phenomenon and the periodic lump wave for the Jimbo–Miwa equation

By using the Hirota bilinear method, new interaction solutions and the periodic lump wave solutions for the Jimbo–Miwa equation are successfully solved via symbolic computation with Maple. These new solutions greatly enrich the existing literature on the Jimbo–Miwa equation. Via the three-dimensional images and density images, the physical characteristics of the interactions and the periodic lump wave are well observed. These physical features of the waves obtained in this paper will be widely used in the fields of electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics.

Dynamics of lump solutions, lump-kink solutions and periodic lump solutions in a (3+1)-dimensional generalized Jimbo–Miwa equation

Under investigation in this paper is a (3+1)-dimensional generalized Jimbo–Miwa equation(gJM), which describes many interesting three dimensional nonlinear waves in physics. Based on the properties of the Bell's polynomial, a very classic method is presented to find the Hirota bilinear form of the equation. By using the resulting bilinear form, we systematically construct its lump solutions and lump-like solutions including lump-kink solutions and periodic lump solutions with their corresponding limitations. Moreover, we show that the conditions satisfying by lump solutions are considered to satisfy several important properties, including localization, positive and analyticity. The interaction solutions among a lump and a kink wave are also provided by considering a special function. The corresponding interaction phenomenon does not possess elasticity, but has a constantly removing process between a lump and a kink wave. Finally, periodic lump solutions are also derived by employing the three-wave method. It is hoped that our results can help rich the dynamic behaviors of the (3+1)-dimensional Jimbo–Miwa-like equations.

Resonant multi-soliton solutions to new (3+1)-dimensional Jimbo–Miwa equations by applying the linear superposition principle

In this paper, the linear superposition principle and Hirota bilinear equations are simultaneously employed to handle two new (3+1)-dimensional Jimbo–Miwa equations. The corresponding resonant multi-soliton solutions and the related wave numbers are formally established, which are totally different from the previously reported ones. Moreover, the extracted N-soliton waves and dispersion relations have distinct physical structures compared to solutions obtained by Wazwaz. Finally, five graphical representations are portrayed by taking definite values to free parameters which demonstrates various versions of traveling solitary waves. The results show the proposed approach provides enough freedom to construct multi-soliton waves that may be related to a large variety of real physical phenomena and, moreover, enriches the solution structure.

All single travelling wave patterns to fractional Jimbo–Miwa equation and Zakharov–Kuznetsov equation

By the complete discrimination system of polynomial method, we obtain the classification and representation of all single travelling wave solutions to \((3+1)\)-dimensional conformal fractional Jimbo–Miwa equation and fractional Zakharov–Kuznetsov equation. These solutions show rich evolution patterns of models described by these two equations.

New periodic wave solutions of a (3+1)-dimensional Jimbo–Miwa equation

A new three-wave method is efficient and well-developed approach to solve nonlinear partial differential equation. In this paper, a (3+1)-dimensional Jimbo–Miwa equation is investigated by using this approach. Some periodic wave solutions and kink solutions are obtained through the Hirota bilinear form. Furthermore, figures of some special periodic wave solutions and kink solutions are presented to illustrate the dynamical features of these solutions.

Bounded and unbounded traveling wave solutions of the (3+1)-dimensional Jimbo-Miwa equation

Abstract This paper studies traveling waves of the (3+1)-dimensional Jimbo-Miwa equation comprehensively and systematically. By transforming its traveling wave system into a dynamical system in R 3 , we apply the bifurcation method of dynamical system to investigate its phase space geometry in detail and obtain the parameter bifurcation sets in which various types of bounded and unbounded traveling waves are identified and simulated. Furthermore, by calculating the complicated elliptic integrals, without any loss, we give exact expressions of all traveling wave solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the bounded and unbounded ones.

Solutions and conservation laws of a generalized second extended (3+1)-dimensional Jimbo-Miwa equation

Abstract In this paper we study a nonlinear multi-dimensional partial differential equation, namely, a generalized second extended (3+1)-dimensional Jimbo-Miwa equation. We perform symmetry reductions of this equation until it reduces to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is obtained in terms of the Weierstrass zeta function. Also travelling wave solutions are derived using the simplest equation method. Finally, the conservation laws of the underlying equation are computed by employing the conservation theorem due to Ibragimov, which include conservation of energy and conservation of momentum laws.

Mixed lump–stripe soliton solutions to a dimensionally reduced generalized Jimbo–Miwa equation

With the help of the direct bilinear method, explicit rational and rational–exponential solutions to a dimensionally reduced generalized Jimbo–Miwa equation are derived by assuming the auxiliary function as the quadratic function and the combinations of the quadratic function and the exponential one. It is shown that three kinds of exact solutions describe the lump, the interaction consisting of one lump and one stripe soliton, and the rogue wave arising from the stripe soliton pair respectively. Local characteristics and interaction properties for these mixed nonlinear waves are analyzed in detail.

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

Based on Hirota bilinear method, four kinds of localized waves, solitons, breathers, lumps and rogue waves of the extended (3+1)-dimensional Jimbo–Miwa equation are constructed. Breathers are obtained through choosing appropriate parameters on soliton solutions, while lumps and rogue waves are derived via the long wave limit on the soliton solutions. The energy, phase shift, shape, and propagation direction of these localized waves can be influenced and controlled by parameters. Considering mixed cases of the above four types of solutions, we also give many kinds of interaction solutions in the same plane with different parameters or different planes with the same parameters. Dynamical characteristics of these localized waves and interaction solutions are further analyzed and vividly demonstrated through figures.

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

In this paper, based on the Hirota bilinear form and a ”rational-cosh-cos” type test function, we will study new interaction solutions among the lump waves, kinky waves and periodic waves for the (3+1)-dimensional Jimbo–Miwa equation. We analyze the dynamic behaviors of the constructed solutions in four special cases in detail. And the dynamic characteristics of the obtained solutions are illustrated by some interesting figures with the help of Maple.

A study of lump-type and interaction solutions to a (3+1)-dimensional Jimbo–Miwa-like equation

Using generalized bilinear equations, we construct a (3+1)-dimensional Jimbo–Miwa-like equation which possesses the same bilinear type as the standard (3+1)-dimensional Jimbo–Miwa equation. Classes of lump-type solutions and interaction solutions between lump-type and kink solutions to the resulting Jimbo–Miwa-like equation are generated through Maple symbolic computation. We discuss the conditions guaranteeing analyticity and positiveness of the solutions. By taking special choices of the involved parameters, 3D plots are presented to illustrate the dynamical features of the solutions.

General lump-type solutions of the (3+1)-dimensional Jimbo–Miwa equation

In this paper, we study the general lump-type solutions of the (3+1)-dimensional Jimbo–Miwa equation via Hirota bilinear method and the ansatz technique. In contrast with lump solutions presented before, we firstly find a general quadratic function solution of the transformed bilinear Jimbo–Miwa equation and then expand it as the sums of squares of linear functions to satisfy analyticity condition. Especially, we get a lump-type solution with fifteen parameters which possess eleven arbitrary independent parameters and four constraint conditions. This solution supplements the existing lump-type solutions obtained previously in the literature. Finally, we conclude that there are only two linearly independent non-constant linear functions in the summation for a positive quadratic function solution.