Negative-order KdV equations in (3+1) dimensions by using the KdV recursion operator

We develop a variety of negative-order Korteweg-de Vries (KdV) equations in (3+1)-dimensions. The recursion operator of the KdV equation is used to derive these higher dimensional models. The new equations give distinct solitons structures and distinct dispersion relations as well. We also determine multiple soliton solutions for each derived model.     

Multisoliton Solutions of the (2+1)-Dimensional KdV Equation

Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2+1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2+1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.     

(2+1)维非线性KdV方程的新孤子结构

通过选取另一类种子解，给出了(2+1)维非线性KdV方程的一类变量分离新解.适当地选择变量分离新解中的任意函数和条件函数，揭示了一类新型孤子结构，如周期性孤波结构、环状孤子结构、曲线型孤子结构等.可以发现(2+1)维非线性KdV方程存在的这类新型孤子结构，是无法通过以往文献中给出的通用变量分离表达式得到的，而且这类新型孤子结构对于实际自然现象的解释有积极的意义. 关键词： 变量分离法 (2+1)维非线性KdV方程 新孤子结构     

New Exact Periodic Solitary-Wave Solutions for New (2+1)-Dimensional KdV Equation

The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions (including kinky periodic solitary-wave solutions, periodic soliton solutions, and crosskink-wave solutions) for the new (2+1)-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensional nonlinear wave field.     

Baeicklund Transformation and Multiple Soliton Solutions for （3＋1）-Dimensional Potential-YTSF Equation

We study an approach to constructing multiple soliton solutions of the (3 1)-dimensional nonlinear evolution equation. We take the (3 1)-dimensional potential- YTSF equation as an example. Using the extended homogeneous balance method, one can find a Backlund transformation to decompose the (3 1)-dimensional potential-YTSF equation into a set of partial differential equations. Starting from these partial differential equations, some multiple soliton solutions for the (3 1)-dimensional potential-YTSF equation are obtained by introducing a class of formal solutions.     

ABUNDANT DROMION-LIKE STRUCTURES TO THE (2+1) DIMENSIONAL KdV EQUATION

The abundant generalized dromion structures for the (2+1)-dimensional KdV equation are obtained using the homogeneous balance method. We give not only the general curve soliton which is finite on a curved line and localized apart from the curve, find but also the dromion solutions which can be driven by two perpendicular line soliton and by two non-perpendicular line soliton and by one line soliton and one curve line soliton. Various types of multi-dromion solutions can be constituted by selecting different arbitrary functions of y. The (1+N) dromion obtained by Radha et al.^[3] is only a very special case of our results.     

A (2+1)-Dimensional Integrable sinh-Gordon Extension

In this letter, we point out that if there is a (2+1)-dimensional extension of the KdV equation,then we can get a corresponding (2+1)-dimensional sine-Gordon (SG) or sinh-Gordon (SHG) extension which is related to the negative flow equation of the KdV extension. The (2+1)-dimensional SG (or SHG) extensions related to the KP, breaking soliton and Nizhnik-Novikov-Veselov equations are known in literature while the (2+1)-dimensional SHG extension related to the negative Aow equation of the Boiti-Leon-Manna-Pempinelli (BLMP) equation is obtained in this letter thanks to the Schwartz form of the BLMP equation being conformal invariant.     

Binary Bell Polynomials,Bilinear Approach to Exact Periodic Wave Solutions of (2+1)-Dimensional Nonlinear Evolution Equations

In the present letter, we get the appropriate bilinear forms of(2+1)-dimensional KdV equation, extended (2+1)-dimensional shallow water wave equation and (2+1)-dimensional Sawada-Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.     

A Series of Exact Solutions for a New （2＋1）-Dimensional Calogero KdV Equation

An algebraic method is proposed to solve a new （2＋1）-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, line soliton solutions, and doubly periodic wave solutions.     

B?cklund transformation and multiple soliton solutions for the (3+1)-dimensional Jimbo-Miwa equation

We study an approach to constructing multiple soliton solutions of the (3+1)-dimensional nonlinear evolution equation. We take the (3+1)-dimensional Jimbo－Miwa (JM) equation as an example. Using the extended homogeneous balance method, one can find a B?cklund transformation to decompose the (3+1)-dimensional JM equation into a linear partial differential equation and two bilinear partial differential equations. Starting from these linear and bilinear partial differential equations, some multiple soliton solutions for the (3+1)-dimensional JM equation are obtained by introducing a class of formal solutions.     

Space-Curved Resonant Line Solitons in a Generalized(2 + 1)-Dimensional Fifth-Order KdV System

On the basis of N-soliton solutions, space-curved resonant line solitons are derived via a new constraint proposed here, for a generalized(2 + 1)-dimensional fifth-order KdV system. The dynamic properties of these new resonant line solitons are studied in detail. We then discuss the interaction between a resonance line soliton and a lump wave in greater detail. Our results highlight the distinctions between the generalized(2+1)-dimensional fifth-order KdV system and the classical type.     

A Series of Variable Separation Solutions and New Soliton Structures of （2＋1）-Dimensional Korteweg-de Vries Equation

Variable separation approach is introduced to solve the （2＋1）-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model （24）. Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.     

Multisoliton Solutions of the (2 1)-Dimensional KdV Equation

Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2 1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2 1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.``     

Novel methods for finding general forms of new multi-soliton solutions to (1+1)-dimensional KdV equation and (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation

In this paper, two novel methods used to solve (1+1) and (2+1)-dimensional completely integrable equations are proposed. The methods are applied to handle the KdV and Kadomtsev–Petviashvili (KP) equations with variable coefficients, and the general forms of new multi-soliton solutions are formally obtained, respectively. In addition, the new multi-soliton solution is suitable to two different type KP equations. Comparing with the Hirota’s method, the results show that new methods are straightforward handling the KdV and KP equations without conjecturing the transformation and good in dealing the equations with variable coefficients.     

A discrete KdV equation hierarchy: continuous limit,diverse exact solutions and their asymptotic state analysis

In this paper, a discrete KdV equation that is related to the famous continuous KdV equation is studied. First, an integrable discrete KdV hierarchy is constructed, from which several new discrete KdV equations are obtained. Second, we correspond the first several discrete equations of this hierarchy to the continuous KdV equation through the continuous limit. Third, the generalized (m, 2N − m)-fold Darboux transformation of the discrete KdV equation is established based on its known Lax pair. Finally, the diverse exact solutions including soliton solutions, rational solutions and mixed solutions on non-zero seed background are obtained by applying the resulting Darboux transformation, and their asymptotic states and physical properties such as amplitude, velocity, phase and energy are analyzed. At the same time, some soliton solutions are numerically simulated to show their dynamic behaviors. The properties and results obtained in this paper may be helpful to understand some physical phenomena described by KdV equations.     

Multiple soliton solutions for the integrable couplings of the KdV and the KP equations

In this work, we study the nonlinear integrable couplings of the KdV and the Kadomtsev-Petviashvili (KP) equations. The simplified Hirota’s method will be used for this study. We show that these couplings possess multiple soliton solutions the same as the multiple soliton solutions of the KdV and the KP equations, but differ only in the coefficients of the transformation used. This difference exhibits soliton solutions for some equations and anti-soliton solutions for others.     

Soliton-Like Solutions for the (2+l)-Dimensional Nonlinear Evolution Equation

By using a homogeneous balance method, we give new soliton-like solutions for the (2+1)-dimensional KdV equation and the (2+1)-dimensional breaking soliton equation. Solitary wave soIutions are shown to be a special case of the present results.     

B(a)cklund Transformation and Multiple Soliton Solutions for (3+1)-Dimensional Potential-YTSF Equation

We study an approach to constructing multiple soliton solutions of the (3 1)-dimensional nonlinear evolu tion equation. We take the (3 1)-dimensional potential-YTSF equation as an example. Using the extended homogeneous balance method, one can find a Backlund transformation to decompose the (3 1)-dimensional potential-YTSF equa tion into a set of partial differential equations. Starting from these partial differential equations, some multiple soliton solutions for the (3 1)-dimensional potential-YTSF equation are obtained by introducing a class of formal solutions.     

Reduction and New Explicit Solutions of (2+1)-Dimensional Breaking Soliton Equation

By applying the Lie group method, the (2+1)-dimensional breaking soliton equation is reduced to some (1+1)-dimensional nonlinear equations. Based upon some new explicit solutions of the (2+1)-dimensional breaking soliton equation are obtained.