Interaction solutions and localized waves to the(2+1)-dimensional Hirota-Satsuma-Ito equation with variable coefficient

This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé analysis.On the basis of the bilinear form,the forms of two-soliton solutions,three-soliton solutions,and four-soliton solutions are studied specifically.The appropriate parameter values are chosen and the corresponding figures are presented.The breather waves solutions,lump solutions,periodic solutions and the interaction of breather waves solutions and soliton solutions,etc.are given.In addition,we also analyze the different effects of the parameters on the figures.The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions.These are important for describing water waves in nature.

Wronskian and Grammian Solutions for Generalized (n + 1)-Dimensional KP Equation with Variable Coefficients

The generalized (n + 1)-dimensional KP equation with variable coefficients is investigated in this paper. The bilinear form of the equation has been obtained by the Hirota direct method. In addition, with the help of Wronskian technique and the Pfaffian properties, Wronskian and Grammian solutions have been generated.

A Generalized Variable-Coefficient Algebraic Method Exactly Solving （3＋1）-Dimensional Kadomtsev-Petviashvilli Equation

A generalized variable-coefficient algebraic method is applied to construct several new families of exact solutions of physical interestfor (3+1)-dimensional Kadomtsev-Petviashvilli (KP) equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.

Painleve Analysis and Determinant Solutions of a (3+1)-Dimensional Variable-Coefficient Kadomtsev-Petviashvili Equation in Wronskian and Grammian Form

在这篇论文，调查集中于一(3+1 ) 维的可变系数的 Kadomtsev Petviashvili (vcKP ) 方程，它能在三种空间尺寸在液体动力学和血浆描述现实主义的非线性的现象。以便学习如此的一个方程的 integrability 性质， Painlev é分析在它上被执行。然后基于截断的 Painlev é扩大，双线性的形式(3+1 ) 维的 vcKP 方程被 Wronskian 技术的优点在某些系数限制下面获得，并且它在 Wronskian 决定因素形式的答案被构造并且验证。除 Wronskian 决定因素答案以外，这被显示出(3+1 ) 维的 vcKP 方程也拥有在 Grammian 决定因素形式的一个答案。

Exact Solutions to a (3+1)-Dimensional Variable-Coefficient Kadomtsev-Petviashvilli Equation via the Bilinear Method and Wronskian Technique

By truncating the Painleve expansion at the constant level term,the Hirota bilinear form is obtainedfor a (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation.Based on its bilinear form,solitary-wavesolutions are constructed via the ε-expansion method and the corresponding graphical analysis is given.Furthermore,the exact solution in the Wronskian form is presented and proved by direct substitution into the bilinear equation.

一个带自相容源的变系数(3+1)维KP方程

运用源生成法构造了一个带自相容源的变系数(3+1)维KP方程,运用Hirota方法对其进行研究,并给出了带自相容源的变系数(3+1)维KP方程的一组贝克隆变换.

一个新型的带自相容源的变系数(3+1)维KP方程

通过引进关于自变量y的任意函数,利用源生成法构造了一个新型的带自相容源的变系数(3+1)维KP方程.

Spatiotemporal Rogue Waves for the Variable-Coefficient (3+1)-Dimensional Nonlinear Schrdinger Equation

With the help of the similarity transformation connected the variable-coefficient (3+1)-dimensional nonlinear Schrdinger equation with the standard nonlinear Schrdinger equation, we firstly obtain first-order and second-order rogue wave solutions. Then, we investigate the controllable behaviors of these rogue waves in the hyperbolic dispersion decreasing profile. Our results indicate that the integral relation between the accumulated time T and the real time t is the basis to realize the control and manipulation of propagation behaviors of rogue waves, such as sustainment and restraint. We can modulate the value T 0 to achieve the sustained and restrained spatiotemporal rogue waves. Moreover, the controllability for position of sustainment and restraint for spatiotemporal rogue waves can also be realized by setting different values of X 0 .

Soliton Solutions, Bcklund Transformations and Lax Pair for a(3 + 1)-Dimensional Variable-Coefficient Kadomtsev–Petviashvili Equation in Fluids

Under investigation in this paper is a(3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation, which describes the propagation of surface and internal water waves. By virtue of the binary Bell polynomials,symbolic computation and auxiliary independent variable, the bilinear forms, soliton solutions, B¨acklund transformations and Lax pair are obtained. Variable coefficients of the equation can affect the solitonic structure, when they are specially chosen, while curved and linear solitons are illustrated. Elastic collisions between/among two and three solitons are discussed, through which the solitons keep their original shapes invariant except for some phase shifts.

Painlevé Analysis, Soliton Collision and B?cklund Transformation for the (3+1)-Dimensional Variable-Coefficient Kadomtsev–Petviashvili Equation in Fluids or Plasmas

In this paper, we investigate a(3+1)-dimensional generalized variable-coefficient Kadomtsev–Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlev′e analysis is performed for us to study the integrability, and we find that the equation is not completely integrable. By virtue of the binary Bell polynomials,bilinear form and soliton solutions are obtained, and B¨acklund transformation in the binary-Bell-polynomial form and bilinear form are derived. Soliton collisions are graphically discussed: the solitons keep their original shapes unchanged after the collision except for the phase shifts. Variable coefficients are seen to affect the motion of solitons: when the variable coefficients are chosen as the constants, solitons keep their directions unchanged during the collision; with the variable coefficients as the functions of the temporal coordinate, the one soliton changes its direction.

变系数(3+1)维破碎孤子方程的复合型解

基于Riccati方程与二阶线性常微分方程构建扩展的辅助方程法,并借助符号计算系统Mathematica,获得了变系数(3+1)维破碎孤子方程的由双曲函数、三角函数和有理函数两两组合的复合型解。

一个混合型的带自相容源的变系数(3+1)维KP方程

通过引入关于变量y,z,t的任意函数,利用源生成法构造了一个混合型的带自相容源的变系数(3+1)-维KP方程.

扩展的同宿检验法与(3+1)维广义KP方程的非行波解

扩展的同宿检验法是求解非线性偏微分方程精确解非常有效的方法.该文将扩展的同宿检验法与分离变量方法相结合,探索(3+1)维广义KP方程的非行波精确解.借助数学软件工具,讨论了4种情形下的非行波精确解.进而,通过分析系数在实数域、复数域中的取值情况,得到了(3+1)维广义KP方程15种类型的精确非行波解.

(3+1)维变系数Burgers方程的类孤子新解

提出函数变换与二阶常系数齐次线性常微分方程相结合的方法,借助符号计算系统Mathematica构造了(3+1)维变系数Burgers方程的类孤子新解,其由指数函数、三角函数和有理函数组成.

(3+1)维变系数Kudryashov⁃Sinelshchikov(K⁃S)方程的同宿呼吸波解和高阶怪波解

基于Hirota双线性方法,利用拓展的同宿呼吸检验法得到了(3+1)维变系数Kudryashov⁃Sinelshchikov(K⁃S)方程的同宿呼吸波解,对该解的参数选取合适的数值,可得到不同结构的同宿呼吸波.通过对同宿呼吸波解的周期取极限,推导出方程的怪波解.最后,构造出一个特殊的高阶多项式作为测试函数,求得该方程的一阶怪波解和二阶怪波解.

变系数(n+1)-维KP方程的Wronskian和Grammian解

基于Hirota直接方法,将变系数(n+1)-维KP方程化成Hirota双线性形式,再借助Wronskian技巧和Pfaffian性质,对该方程进行求解,得到了其广义的Wronskian解和Grammian解.

变系数(3+1)维ZK方程的精确类孤子解

本文对标准椭圆方程的解进行分类且给出所有独立解.利用这些解并借助Mathematica系统获得了变系数(3+1)维ZK方程的多个类孤子解,包括指数函数解,周期函数解,双曲函数解,双周期雅可比椭圆函数解,双周期Weierstrass椭圆形式解以及有理解.

(3+1)维Boiti-Leon-Manna-Pempinelli方程的新精确解

本文主要利用扩展的G′/G展开法,由齐次平衡和辅助方程,提出各种不同形式的测试函数,特别是提出变系数测试函数,对经典的(3+1)维Boiti-Leon-Manna-Pempinelli(BLMP)方程和修正的BLMP方程进行计算和分析,得到相应的新精确解。这些解主要包含了双曲函数、三角函数和有理函数的变系数形式等。通过对新解中的自由参数和变系数进行分析和赋值,得到相应解的图形,并解释在不同条件下的非线性自然现象。

Resonant multiple wave solutions to some integrable soliton equations

To transform the exponential traveling wave solutions to bilinear differential equations, a sufficient and necessary condition is proposed. Motivated by the condition, we extend the results to the(2+1)-dimensional Kadomtsev–Petviashvili(KP) equation, the(3+1)-dimensional generalized Kadomtsev–Petviashvili(g-KP) equation, and the B-type Kadomtsev–Petviashvili(BKP) equation. Aa a result, we obtain some new resonant multiple wave solutions through the parameterization for wave numbers and frequencies via some linear combinations of exponential traveling waves. Finally, these new resonant type solutions can be displayed in graphs to illustrate the resonant behaviors of multiple wave solutions.