Miwa Equation Research Articles

Lie symmetry analysis, optimal system, and generalized group invariant solutions of the (2 + 1)‐dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)‐dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one‐dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.

Applications of Homogenous Balance Principles Combined with Fractional Calculus Approach and Separate Variable Method on Investigating Exact Solutions to Multidimensional Fractional Nonlinear PDEs

We investigate the exact solutions of multidimensional time-fractional nonlinear PDEs (fnPDEs) in this paper. In terms of the fractional calculus properties and the separate variable method, we present a new homogenous balance principle (HBP) on the basis of the (1 + 1)-dimensional time fnPDEs. Taking advantage of the new types of HBP together with fractional calculus formulas that subtly avoid the chain rule, the fnPDEs can be reduced to spatial PDEs, and then we solve these PDEs by the fractional calculus method and the separate variable approach. In this way, some new type exact solutions of the certain time-fractional (2 + 1)-dimensional KP equation, (3 + 1)-dimensional Zakharov–Kuznetsov (ZK) equation, and Jimbo–Miwa (JM) equation are explicitly obtained under both Riemann–Liouville derivatives and Caputo derivatives. The dynamical analysis of solutions is shown by numerical simulations of taking property parameters as well.

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.

Construction of abundant solutions of the (2+1)-dimensional time-dependent Date–Jimbo–Kashiwara–Miwa equation

Investigated in the current paper is the time-dependent Date–Jimbo–Kashiwara–Miwa equation in (2+1)-dimensions which works as a model for describing the propagation of nonlinear dispersive waves in inhomogeneous media. By implementing the homoclinic test method, the breather-kink wave solution to the considered equation is first generated. Then, based on the resulting solution and a parameter transformation, the double-solitary wave solution is obtained. Finally, the rogue wave emerges by taking a limit behavior. Meanwhile, some graphs are made to shed light on the structural characteristics.

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.

A (2+1)-dimensional time-dependent Date–Jimbo–Kashiwara–Miwa equation: Painlevé integrability and multiple soliton solutions

In this work, we extend the (2+1)-dimensional Date- Date–Jimbo–Kashiwara–Miwa equation to a new version with time-dependent coefficients. The integrability property of this developed equation is demonstrated by exhibiting the Painlevé test. We show that this time-dependent equation admits multiple real and multiple complex soliton solutions. Moreover, we derive other solitonic and periodic solutions.

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.

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.

Pfaffians of B-type Kadomtsev–Petviashvili equation and complexitons to a class of nonlinear partial differential equations in (3 $$+$$ + 1) dimensions

The aim of this paper is to investigate a class of generalised Kadomtsev–Petviashvili (KP) and B-type Kadomtsev–Petviashvili (BKP) equations, which include many important nonlinear evolution equations as its special cases. By applying the fundamental Pfaffian identity, a general Pfaffian formulation is established and all the involved generating functions for Pfaffian entries need to satisfy a system of combined linear partial differential equations. The illustrative examples of the presented Pfaffian solutions are given for the (3 $$+$$ 1)-dimensional generalised KP, Jimbo–Miwa and BKP equations. Moreover, we use the linear superposition principle to generate exponential travelling wave solutions and mixed resonant solutions of the considered equations.

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.

Grammian solutions for a (2+1)-dimensional integrable coupled modified Date–Jimbo–Kashiwara–Miwa equation

In this paper, Grammian solutions to a (2[Formula: see text]+[Formula: see text]1)-dimensional modified Date–Jimbo–Kashiwara–Miwa (mDJKM) equation are presented by using Hirota bilinear method and perturbation expansion. Starting from the Grammian solutions, an integrable coupled mDJKM equation is then obtained and the corresponding Grammian solutions are first derived by utilizing the source generation procedure. Besides, we also construct and solve a coupled DJKM equation via source generation procedure. It is interesting that the coupled mDJKM system constitute a bilinear Bäcklund transformation for the coupled DJKM system. This means that the commutativity of source generation procedure and Bäcklund transformation is valid for the (2[Formula: see text]+[Formula: see text]1)-dimensional DJKM equation.

Interaction solutions between lump and stripe soliton to the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this paper, we apply the ansatz method to the multi-linear form of the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation for constructing interaction solutions. By taking the ansatz as the quadratic function or the linear combination of the quadratic function and the exponential one, explicit rational and rational-exponential solutions are derived. It is shown that these exact solutions describe the lump, the lump–stripe soliton interaction with fission and fusion phenomena, and a rogue wave excited from the stripe soliton pair, respectively.

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.