Dimensional Jimbo Miwa Equation Research Articles

Classifications on the travelling wave solutions to the (3+1)-dimensional generalized KP and Jimbo-Miwa equations

In this paper, we use the powerful tan(F (ξ) / 2) -expansion method with the help of Wolfram Mathematica 9 to investigate the solution structures of two well-known nonlinear evolution equations, namely ; the (3+1)-dimensional generalized KP and the (3+1)-dimensional Jimbo-Miwa equations. We obtain new solutions such as hyperbolic function, trigonometric function, exponential function and rational function solutions. We plot two- and three-dimensional graphics of some obtained results using the same program, Wolfram Mathematica 9.

Hybrid Solutions of (3 + 1)-Dimensional Jimbo-Miwa Equation

The rational solutions, semirational solutions, and their interactions to the (3+1)-dimensional Jimbo-Miwa equation are obtained by the Hirota bilinear method and long wave limit. The hybrid solutions contain rogue wave, lump solution, and the breather solution, in which the breathers which are manifested as growing and decaying periodic line waves show different dynamics in different planes. Rogue waves are localized in time and are obtained theoretically as a long wave limit of breathers with indefinitely larger periods; they arise from a constant background at t≪0 and then disappear in the constant background when time goes on. More importantly, the interactions between some hybrid solutions are demonstrated in detail by the three-dimensional figures, such as hybrid solution between the stripe soliton and breather and hybrid solution between stripe soliton and lump solution.

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.

Abundant lump-type solutions of the Jimbo–Miwa equation in ([formula omitted])-dimensions

Based on the Hirota bilinear form of the (3+1)-dimensional Jimbo–Miwa equation, ten classes of its lump-type solutions are generated via Maple symbolic computations, whose analyticity can be easily achieved by taking special choices of the involved parameters. Those solutions supplement the existing lump-type solutions presented previously in the literature.

Lump-Type Solutions to the (3+1)-Dimensional Jimbo-Miwa Equation

Abstract: Taking advantage of the Hirota bilinear form, four classes of lump-type solutions to the (3+1)-dimensional Jimbo-Miwa equation are presented through symbolic computation with Maple. Special choices of the involved parameters guaranteeing analyticity of the fourth solution are given, together with two particular lump-type solutions.

Multiple-soliton solutions for extended (3+1)-dimensional Jimbo–Miwa equations

In this work we investigate two extended (3+1)-dimensional Jimbo–Miwa equations. We use the simplified Hirota’s method to derive multiple soliton solutions of distinct physical structures for each extended equation. We show that the dispersion relations and the phase shifts of the extended equations are distinct compared to the dispersion and shifts of the Jimbo–Miwa equation.

Bäcklund transformations, Lax system, conservation laws and multisoliton solutions for Jimbo–Miwa equation with Bell-polynomials

Based on binary Bell polynomial approach, the bilinear equation and Ba¨cklund transformations for (3+1)-dimensional Jimbo–Miwa equation are obtained. By virtue of Cole–Hopf transformation, Lax system is constructed by direct linearization of coupled system of binary Bell polynomials. Furthermore, infinite conservation laws are obtained from two field condition in quick and natural way. Finally, a test function of extended three wave method is used to construct multisoliton solutions via bilinear equation.

The Rational Solutions and Quasi-Periodic Wave Solutions as well as Interactions ofN-Soliton Solutions for 3 + 1 Dimensional Jimbo-Miwa Equation

The exact rational solutions, quasi-periodic wave solutions, andN-soliton solutions of 3 + 1 dimensional Jimbo-Miwa equation are acquired, respectively, by using the Hirota method, whereafter the rational solutions are also called algebraic solitary waves solutions and used to describe the squall lines phenomenon and explained possible formation mechanism of the rainstorm formation which occur in the atmosphere, so the study on the rational solutions of soliton equations has potential application value in the atmosphere field; the soliton fission and fusion are described based on the resonant solution which is a special form of theN-soliton solutions. At last, the interactions of the solitons are shown with the aid ofN-soliton solutions.

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

In this paper, we have proposed new test function to construct exact solitary wave solutions for nonlinear evolution equations. The (3+1)-dimensional Jimbo–Miwa equation is taken as an example to illustrate the effectiveness of proposed test function. With aid of Maple program, cumbersome calculation is carried out to obtain new exact solutions for said equation. The exact solutions obtained enrich the solution structure of higher-dimensional nonlinear wave equation. Evidently, this method can be applied to other nonlinear evolution equations as well.

New soliton solutions to the (3+1)-dimensional Jimbo–Miwa equation

In this paper by using the sub equation method we obtained new analytical solutions, such as trigonometric function, rational function, and hyperbolic function solutions for the (3+1)-dimensional Jimbo–Miwa equation [1](1)uxxx+3uxuxy+3uyuxx+2uyt−3uxz=0.Under some parameter conditions, exact solitary wave solutions are obtained. The results can be compared with other methods used to solve the same model.

Cross-kink multi-soliton solutions for the (3+1)-D Jimbo-Miwa equation

Purpose – The purpose of this paper is to reveal the dynamical behavior of higher dimensional nonlinear wave by searching for the multi-wave solutions to the (3+1)-D Jimbo-Miwa equation. Design/methodology/approach – The authors apply bilinear form and extended homoclinic test approach to the (3+1)-D Jimbo-Miwa equation. Findings – In this paper, by using bilinear form and extended homoclinic test approach, the authors obtain new cross-kink multi-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the periodic breathertype of kink three-soliton solutions, the cross-kink four-soliton solutions, the doubly periodic breather-type of soliton solutions and the doubly periodic breather-type of cross-kink two-soliton solutions. It is shown that the extended homoclinic test approach, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving higher dimensional nonlinear evolution equations in mathematical physics. Research limitations/implications – The research manifests that the structures of the solution to higher dimensional nonlinear equations are diversified and complicated. Originality/value – The methods used in this paper can be widely applied to the research of spatial and temporal characteristics of nonlinear equations in physics and engineering technology. These methods are also conducive for people to know objective laws and grasp the essential features of the development of the world.

Bilinear Form and Two Bäcklund Transformations for the (3+1)-Dimensional Jimbo-Miwa Equation

With Bell polynomials and symbolic computation, this paper investigates the (3+1)-dimensional Jimbo-Miwa equation, which is one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems. We derive a bilinear form and construct a bilinear Bäcklund transformation (BT) for the (3+1)-dimensional Jimbo-Miwa equation, by virtue of which the soliton solutions are obtained. Bell-polynomial-typed BT is also constructed and cast into the bilinear BT.

New solitary wave solutions for two nonlinear evolution equations

In this paper, multiple exp-function method is employed to investigate exact multiple wave solutions for (2+1)-dimensional potential Kadomtsev–Petviashvili equation and (3+1)-dimensional Jimbo–Miwa equation. Not only already known multiple wave solutions are recovered, but also several new or more general multiple wave solutions are obtained.

New Extended (G’/G)-Expansion Method and its Application in the (3+1)-Dimensional Equation to Find New Exact Traveling Wave Solutions

A new extended (G ' =G)-expansion method is presented in this paper to construct more general type and new traveling wave solutions of nonlinear partial differential equations. To illustrate the novelty and advantage of the proposed method, we solve the (3+1)-dimensional Jimbo-Miwa equation. Abundant exact traveling wave solutions of this equation is ob- tained, which successfully recover most of the previously published solutions. Many of those solutions are found for the first time. Furthermore, the results reveal that the proposed method is very elementary, effective and can be used for many other nonlinear partial differential equations.

Traveling Wave Solutions and Infinite-Dimensional Linear Spaces of Multiwave Solutions to Jimbo-Miwa Equation

The traveling wave solutions and multiwave solutions to (3 + 1)-dimensional Jimbo-Miwa equation are investigated in this paper. As a result, besides the exact bounded solitary wave solutions, we obtain the existence of two families of bounded periodic traveling wave solutions and their implicit formulas by analysis of phase portrait of the corresponding traveling wave system. We derive the exact 2-wave solutions and two families of arbitrary finiteN-wave solutions by studying the linear space of its Hirota bilinear equation, which confirms that the (3 + 1)-dimensional Jimbo-Miwa equation admits multiwave solutions of any order and is completely integrable.

English

The functional variable method is a powerful mathematical tool for obtaining exact solutions of nonlinear evolution equations in mathematical physics. In this paper, the functional variable method is used to establish exact solutions of the (2+1)-dimensional Kadomtsov-Petviashivilli-Benjamin-Bona-Mahony (KP-BBM) equation, the (2+1)-dimensional Konopelchenko-Dubrovsky equation, the (3+1)-dimensional Burgers equation and the (3+1)- dimensional Jimbo-Miwa equation. The exact solutions of these four nonlinear equations including solitary wave solutions and periodic wave solutions are obtained. It is shown that the proposed method is effective and can be applied to many other nonlinear evolution equations. Comparison between our results obtained in this paper and the well-known results obtained by different authors using different methods are presented. Key words: Functional variable method, nonlinear evolution equations, exact solutions, solitary wave solutions, Periodic wave solutions.

Meromorphic solutions of an auxiliary ordinary differential equation using complex method

In this paper, we employ the complex method to obtain first all meromorphic solutions of an auxiliary ordinary differential equation and then find all meromorphic exact solutions of the classical Korteweg–de Vries equation, Boussinesq equation, ( 3 + 1)‐dimensional Jimbo–Miwa equation, and Benjamin–Bona–Mahony equation. Our results show that the method is more simple than other methods. Copyright © 2013 John Wiley & Sons, Ltd.

Pfaffian solutions and extended Pfaffian solutions to (3 + 1)-dimensional Jimbo–Miwa equation

Based on the Pfaffian derivative formula and Hirota bilinear method, the Pfaffian solutions to (3+1)-dimensional Jimbo–Miwa equation are obtained under a set of linear partial differential condition. Moreover, we extend the linear partial differential condition and proved that (3+1)-dimensional Jimbo–Miwa equation has extended Pfaffian solutions. As examples, special exact two-soliton solution and three-soliton solution are computed and plotted. Our results show that (3+1)-dimensional Jimbo–Miwa equation has Pfaffian solutions like BKP equation.

Wronskian and Grammian solutions for the (3+1)-dimensional Jimbo—Miwa equation

In this paper, based on Hirota's bilinear method, the Wronskian and Grammian techniques, as well as several properties of the determinant, a broad set of sufficient conditions consisting of systems of linear partial differential equations are presented. They guarantee that the Wronskian determinant and the Grammian determinant solve the (3 + 1)-dimensional Jimbo—Miwa equation in the bilinear form. Then some special exact Wronskian and Grammian solutions are obtained by solving the differential conditions. At last, with the aid of Maple, some of these special exact solutions are shown graphically.

Generalized Wronskian solutions for the (3 + 1)-dimensional Jimbo–Miwa equation

More general Wronskian conditions are constructed for the (3+1)-dimensional Jimbo–Miwa equation. As a result, the generalized Wronskian solutions including rational solutions, positons, negatons, solitons and interaction solutions are obtained. Moreover, some new solutions, together with some known solutions given by other solving methods, are derived from these Wronskian solutions.