Interactions among special embed-solitons for the (3+1)-dimensional Burgers equation

With the help of a modified mapping method and a new mapping method, we re-study the （3＋1）-dimensional Burgers equation, and derive two families of variable separation solutions. By selecting appropriate functions in the variable separation solution, we discuss the interaction behaviors among taper-like, plateau-type rings, and rectangle-type embed-solitons in the periodic wave background. All the interaction behaviors are completely elastic, and no phase shift appears after interaction.

Embed-Solitons and Their Evolutional Behaviors of (3+1)-Dimensional Burgers System

With the help of an extended mapping approach and a linear variable separation method,new families ofvariable separation solutions with arbitrary functions for the(3+1)-dimensional Burgers system are derived.Based onthe derived exact solutions, some novel and interesting localized coherent excitations such as embed-solitons are revealedby selecting appropriate boundary conditions and/or initial qualifications.The time evolutional properties of the novellocalized excitation are also briefly investigated.

Lump and interaction solutions to the (3+1)-dimensional Burgers equation

The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.

Embedded-Soliton and Complex Wave Excitations of (3+1)-Dimensional Burgers System

从扩大印射的途径和一个线性可变分离方法开始，我们与一些任意的函数发现可变分离解决方案的新家庭为(3+1 ) 维的汉堡包系统。然后基于象 embedded-solitons 那样的导出的准确答案，某小说和有趣的局部性的协调刺激，象逐渐变细一样 soliton，在周期的波浪背景的复杂波浪刺激被介绍适当边界条件或起始的资格揭示。复杂波浪刺激的进化性质简短被调查。

New Exact Solutions and Localized Structures for (3+1)-Dimensional Burgers System

与一条扩大印射的途径和一个线性可变分离方法，有任意的函数为的可变分离解决方案(包括的独居的波浪解决方案，周期的波浪解决方案，和合理函数解决方案) 的新家庭(3+1 ) 维的汉堡包系统被导出。基于导出的刺激，我们获得一些新奇局部性的协调结构并且学习在 solitons 之间的相互作用。

Lie symmetry analysis and invariant solutions for the(3+1)-dimensional Virasoro integrable model

Lie symmetry analysis is applied to a(3+1)-dimensional Virasoro integrable model and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with arbitrary functions for the(3+1)-dimensional Virasoro integrable model,including the interaction solution between a kink and a soliton,the lump-type solution and periodic solutions,have been studied analytically and graphically.

Interaction solutions for the second extended(3+1)-dimensional Jimbo–Miwa equation

Based on the Hirota bilinear method,the second extended(3+1)-dimensional Jimbo–Miwa equation is established.By Maple symbolic calculation,lump and lump-kink soliton solutions are obtained.The interaction solutions between the lump and multi-kink soliton,and the interaction between the lump and triangular periodic soliton are derived by combining a multi-exponential function or trigonometric sine and cosine functions with quadratic functions.Furthermore,periodiclump wave solution is derived via the ansatz including hyperbolic and trigonometric functions.Finally,3D plots,2D curves,density plots,and contour plots with particular choices of the suitable parameters are depicted to illustrate the dynamical features of these solutions.

New periodic wave solutions, localized excitations and their interaction for (2+1)-dimensional Burgers equation

Based on the B/icklund method and the multilinear variable separation approach （MLVSA）, this paper finds a general solution including two arbitrary functions for the （2＋1）-dimensional Burgers equations. Then a class of new doubly periodic wave solutions for （2＋l）-dimensional Burgers equations is obtained by introducing appropriate Jacobi elliptic functions, Weierstrass elliptic functions and their combination in the general solutions （which contains two arbitrary functions）. Two types of limit cases are considered. Firstly, taking one of the moduli to be unity and the other zero, it obtains particular wave （called semi-localized） patterns, which is periodic in one direction, but localized in the other direction. Secondly, if both moduli are tending to 1 as a limit, it derives some novel localized excitations （two-dromion solution）.

(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Burgers方程的对称约化、精确解和守恒律(英文)

应用改进的CK直接方法得到了(3+1)维Burgers方程的对称以及新旧解之间的关系,并由此得到方程部分新的显示解.最后利用对称和守恒律之间的密切关系,得出了此方程的守恒律.

3+1维Burgers方程的Painlevé性质和Bcklund变换及严格解

3+1维的Burgers方程是物理学的重要方程之一.利用奇性分析方法证明了3+1维Burgers方程的Painlevé性质;然后,利用截断的Painlevé展开给出了3+1维Burgers方程的Bcklund变换;最后,由简单的特解出发,利用贝克隆变换得到了3+1维Burgers方程的大量新解.

(3+1)维非线性Burgers系统新的局域激发结构

在(3+1)维非线性Burgers系统分离变量解的基础上,借助于数学软件Mathematica进行数值模拟,得到了系统的新的丰富的局域激发结构。结果表明,应用扩展的Riccati方程映射法得到的高维非线性系统的解具有丰富的局域激发结构。

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrdinger equation with variable coefficients

In this paper, the extended symmetry transformation of （3＋1）-dimensional （3D） generalized nonlinear Schrodinger （NLS） equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.

A more general form of lump solution,lumpoff,and instanton/rogue wave solutions of a reduced (3+1)-dimensional nonlinear evolution equation

In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.

Symmetry analysis and explicit solutions of the (3+1)-dimensional baroclinic potential vorticity equation

This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding （2＋1）-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.

New exact solutions of a (3+1)-dimensional Jimbo-Miwa system

By using the （G＇/G）-expansion method and the variable separation method, a new family of exact solutions of the （3＋1）-dimensional Jimbo-Miwa system is obtained. Based on the derived solitary wave solutions, we obtain some special localized excitations and study the interactions between two solitary waves of the system.

Wronskian Determinant Solutions for the (3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation

In this paper, we consider (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Based on the bilinear form, we derive exact solutions of (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation by using the Wronskian technique, which include rational solutions, soliton solutions, positons and negatons.

Classification of All Single Traveling Wave Solutions to (3 + 1)-Dimensional Breaking Soliton Equation

In order to get the exact traveling wave solutions to nonlinear partial differential equation, the complete discrimination system for polynomial and direct integral method are applied to the considered equation. All single traveling wave solutions to the equation can be obtained. As an example, we give the solutions to (3 + 1)-dimensional breaking soliton equation.

Rogue waves of a(3+1)-dimensional BKP equation

We investigate certain rogue waves of a(3+1)-dimensional BKP equation via the Kadomtsev-Petviashili hierarchy reduction method.We obtain semi-rational solutions in the determinant form,which contain two special interactions:(i)one lump develops from a kink soliton and then fuses into the other kink one;(ii)a line rogue wave arises from the segment between two kink solitons and then disappears quickly.We find that such a lump or line rogue wave only survives in a short time and localizes in both space and time,which performs like a rogue wave.Furthermore,the higher-order semi-rational solutions describing the interaction between two lumps(one line rogue wave)and three kink solitons are presented.