Miwa Equation Research Articles

Complexiton solutions and soliton solutions: $$(2+1)$$ ( 2 + 1 ) -dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, we derive the complexiton solutions for Date–Jimbo–Kashiwara–Miwa (DJKM) equation using the extended transformed rational function algorithm that relies on the Hirota bilinear form of the considered equation. Additional solutions such as complex-valued solutions also fall out of this integration scheme. Multisoliton-type solutions, in other words one-soliton, two-soliton and three-soliton solutions, which comprise both wave frequencies and generic phase shifts are presented through the medium of the multiple exp-function methodology which falls out as a result of generalisation of Hirota’s perturbation technique.

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.

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.

Exact solitary wave solutions for two nonlinear systems

In this paper, we employ the powerful sine-Gordon expansion method in investigating the solitary wave solutions of the fifth-order nonlinear equation and the Date–Jimbo–Kashiwara–Miwa equation with symbolic computation. We obtain the hyperbolic, trigonometric and complex solutions and the corresponding plots of the solitary wave solutions are given out analytically and graphically.

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.

Novel solitary wave solutions for the [formula omitted]-dimensional extended Jimbo–Miwa equations

This paper retrieves new periodic solitary wave solutions for the (3+1)-dimensional extended Jimbo–Miwa equations, based on the Hirota bilinear method, by utilizing Maple software. As a result, the Hirota bilinear method is successfully employed and acquired several classes of solitary wave solutions in terms of a new combination of exponential function, trigonometric function and hyperbolic functions. All solutions have been verified back into its corresponding equation by Maple. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, it has been illustrated that the executed method is robust and more efficient than other methods and the obtained solutions are trustworthy in the applied sciences.

On Painlevé Analysis, Symmetry Group and Conservation Laws of Date–Jimbo–Kashiwara–Miwa Equation

In this study, the Painleve property of Date–Jimbo–Kashiwara–Miwa equation is verified with WTC approach and the exact soliton solution is obtained using Pickering’s algorithm. We find the Lie point symmetries and three independent similarity reductions. Further, two different techniques are employed independently to construct conservation laws. Multipliers in the direct method have given the infinite number of non-trivial local conservation laws and self-adjointness of equation helped us to construct infinite local conservation laws associated with Lie symmetries. The existence of the infinite number of local conservation laws strongly justifies the complete integrability of equation.

Meromorphic exact solutions of two extended (3+1)-dimensional Jimbo–Miwa equations

In this work, abundant meromorphic exact solutions of two kinds of extended (3+1)-dimensional Jimbo–Miwa equations are obtained by making use of the complex method, which contain rational solutions, exponential function solutions and elliptic function solutions. The properties of the results are also shown by some figures.

Interaction solutions for a reduced extended $$\mathbf{(3}\varvec{+}{} \mathbf{1)}$$(3+1)-dimensional Jimbo–Miwa equation

In this paper, the exact solutions of a reduced extended $$(3+1)$$ -dimensional Jimbo–Miwa equation are investigated with the help of its bilinear representation and symbolic computation. Firstly, a kind of bright–dark lump wave solutions is directly obtained by taking the solution F in bilinear equation as a quadratic function. Furthermore, the interaction solutions between one lump wave and one stripe wave are also presented by taking F as a combination of quadratic function and exponential function. Finally, by taking F as a combination of quadratic function and hyperbolic cosine function, the rogue wave which aroused by the interaction between lump soliton and a pair of stripe solitons are obtained. The dynamic properties of the above three kinds of exact solutions are displayed vividly by figures.

Some integrable maps and their Hirota bilinear forms

We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota–Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case.

Bäcklund transformation, infinitely-many conservation laws, solitary and periodic waves of an extended (3 + 1)-dimensional Jimbo–Miwa equation with time-dependent coefficients

In this paper, an extended (3+1)-dimensional Jimbo–Miwa equation with time-dependent coefficients is investigated, which comes from the second member of the Kadomtsev–Petviashvili hierarchy and is shown to be conditionally integrable. Bilinear form, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived via the binary Bell polynomials and symbolic computation. With the help of the bilinear form, one-, two- and three-soliton solutions are obtained via the Hirota method, one-periodic wave solutions are constructed via the Riemann theta function. Additionally, propagation and interaction of the solitons are investigated analytically and graphically, from which we find that the interaction between the solitons is elastic and the time-dependent coefficients can affect the soliton velocities, but the soliton amplitudes remain unchanged. One-periodic waves approach the one-solitary waves with the amplitudes vanishing and can be viewed as a superposition of the overlapping solitary waves, placed one period apart.

A KdV-Type Wronskian Formulation to Generalized KP, BKP and Jimbo–Miwa Equations**Supported by the National Natural Science Foundation of China under Grant No. 11371326

The purpose of this paper is to introduce a class of generalized nonlinear evolution equations, which can be widely applied to describing a variety of phenomena in nonlinear physical science. A KdV-type Wronskian formulation is constructed by employing the Wronskian conditions of the KdV equation. Applications are made for the (3+1)-dimensional generalized KP, BKP and Jimbo–Miwa equations, thereby presenting their Wronskian sufficient conditions. An N-soliton solution in terms of Wronskian determinant is obtained. Under a dimensional reduction, our results yield Wronskian solutions of the KdV equation.

Wronskian and Grammian solutions for a [formula omitted]-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this paper, a (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation is investigated. Based on the Hirota method and auxiliary variable, Bäcklund transformation is obtained. Under certain conditions, the Hirota bilinear forms can be reduced to the Plücker and Jacobi identities via the Wronskian and Grammian solutions. Through the Wronskian technique and Pfaffian derivative formulae, N-soliton solutions in the Wronskian and Grammian are given. Graphically presented, soliton collisions are elastic both in the Wronskian and Grammian, and after each collision, the solitons keep their amplitudes unchanged except for the phase shifts.

Rogue wave and a pair of resonance stripe solitons to a reduced (3+1)-dimensional Jimbo–Miwa equation

In this paper, a combination of stripe soliton and lump soliton is discussed to a reduced (3+1)-dimensional Jimbo–Miwa equation, in which such solution gives rise to two different excitation phenomena: fusion and fission. Particularly, a new combination of positive quadratic functions and hyperbolic functions is considered, and then a novel nonlinear phenomenon is explored. Via this method, a pair of resonance kink stripe solitons and rogue wave is studied. Rogue wave is triggered by the interaction between lump soliton and a pair of resonance kink stripe solitons. It is exciting that rogue wave must be attached to the stripe solitons from its appearing to disappearing. The whole progress is completely symmetry, the rogue wave starts itself from one stripe soliton and lose itself in another stripe soliton. The dynamic properties of the interaction between one stripe soliton and lump soliton, rogue wave are discussed by choosing appropriate parameters.

Lump and lump–kink solutions of the (3+1)-dimensional Jimbo–Miwa and two extended Jimbo–Miwa equations

In this paper, we study the lump solutions and their dynamics of the ( 3 + 1 ) -dimensional Jimbo–Miwa equation and two extended Jimbo–Miwa equations via bilinear forms. Based on the obtained lump solutions, the lump–kink solution which contains interaction between a lump and a kink wave of the ( 3 + 1 ) -dimensional Jimbo–Miwa equation is also obtained.

Construction of exact solutions to the space–time fractional differential equations via new approach

In the present article, the local fractional derivatives and the exp(−Φ(ξ)) method are used to construct the exact solutions of nonlinear space–time fractional partial differential equations. For illustrating the validity of the method, it is applied to the space–time fractional (3+1)-dimensional nonlinear Jimbo–Miwa equation and nonlinear Hirota–Satsuma coupled KdV system. This approach is an efficient mathematical tool for solving fractional differential equations and it can be applied to other nonlinear fractional differential equations.