(2+1)维 Zakharov-Kuznetsov 方程的精确解和孤子结构

在符号计算软件Maple的帮助下,利用改进的Riccati方程映射法得到了(2+1)维Zakharov-Kuznetsov方程(ZK)的新显式精确解. 根据得到的解,研究了ZK方程的特殊孤子结构. 关键词： 改进的Riccati方程映射法 Zakharov-Kuznetsov方程 精确解 孤子结构     

(3+1)维Nizhnik-Novikov-Veselov方程的孤子解和周期解

采用行波法约化方程,建立一种变换关系,把求解(3+1)维NizhnikNovikovVeselov(NNV)方程的解转化为求解一维非线性KleinGordon方程的解,从而得到了(3+1)维NNV方程的孤子解和周期解. 关键词： (3+1)维Nizhnik-Novikov-Veselov方程 非线性Klein-Gordon方程 孤子解 周期解     

(2+1)-维Wick类型随机广义KP方程的类孤子解

利用埃尔米特变换和特殊的截断展开法求出(2+1)-维Wick类型随机广义KP方程的类孤子解. 这种方法的基本思想是通过埃尔米特变换把(2+1)-维Wick类型随机广义KP方程变成的(2+1)-维广义变系数KP方程,利用特殊的截断展开方法求出方程的解,然后通过埃尔米特的逆变换求出方程的随机解.     

在Jacobi正弦周期波背景下的dromion孤立波及其演化

利用改进的Riccati方程映射法、Bcklund变换法和变量分离法，得到了(2+1)维广义Breor-Kaup(GBK)方程的新显式精确解.根据得到的解，找到了GBK方程的复合波，并进一步研究了在Jacobi正弦周期波背景下dromion孤立波的演化. 关键词： 改进的映射法 广义Breor-Kaup方程 复合波 Jacobi正弦周期波     

(3+1)维Jimbo-Miwa方程的精确解及局域激发

在符号计算软件 Maple 的帮助下,利用投射方程法和变量分离法,得到了(3+1)维 Jimbo-Miwa(JM)方程的新显式精确解. 根据得到的孤立波解,研究了 JM 方程新颖的局域激发. 关键词： 投射方程法 Jimbo-Miwa方程 精确解 局域激发     

(2+1)维耗散长波方程的变量分离解之注记

对文献给出的(2+1)维耗散长波方程的变量分离解作了进一步的研究,首先说明拓展的F/G展开法得到的解等价于拓展的tanh展开法得到的解;然后通过一个变换以及修正的齐次平衡法,给出了(2+1)维耗散长波方程更多的精确解.     

(2+1)维耗散长波方程的变量分离解

借助Mathematica符号计算软件,利用拓展的F/G展开法和变量分离法,得到(2+1)维耗散长波方程的精确解.通过选择适当的函数,获得(2+1)维耗散长波方程的亮暗dromion解和周期孤波解.     

一类非线性方程的compacton解及其移动compacton解

研究一类非线性方程,即广义Camassa-Holm方程C(n):ut+kux+β1u\{xxt\}+β2u\{n+1\}x+β3uxun\{xx\}+β4uun\{xxx\}=0.通过四种拟设得到丰富的精确解，特别是当k≠0时得到了com pacton解,当k=0时得到了移动compacton解.最后利用线 性化的方法得到了其他形式的广义Camassa-Holm方程的compacton解. 关键词： 广义Camassa-Holm方程 compacton解 移动compacton解     

(3+1)维Burgers系统的新精确解及其特殊孤子结构

将改进的Riccati方程映射法和变量分离法推广到(3+1)维Burgers系统,得到了该系统的新显式精确解.根据得到的孤波解,构造出Burgers系统的几种特殊孤子结构,例如柱状孤子、状孤子和内嵌孤子等,研究了孤子间的相互作用. 关键词： 改进的映射法 (3+1)维Burgers系统 孤子结构 相互作用     

(3+1)维非线性Burgers系统的新的分离变量解及其局域激发结构与分形结构

将扩展的Riccati方程映射法推广到了(3+1)维非线性Burgers系统，得到了系统的分离变量解；由于在解中含有一个关于自变量(x,y,z,t)的任意函数，通过对这个任意函数的适当选取，并借助于数学软件Mathematica进行数值模拟，得到了系统的新而丰富的局域激发结构和分形结构.结果表明，扩展的Riccati方程映射法在求解高维非线性系统时，仍然是一种行之有效的方法，并且可以得到比(2+1)维非线性系统更为丰富的局域激发结构. 关键词： 扩展的Riccati方程映射法 (3+1)维非线性Burgers方程 局域激发结构 分形结构     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.     

Symmetry Analysis and Conservation Laws for the Hunter-Saxton Equation

In this paper,the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation(HSE) is analyzed.By applying the basic Lie symmetry method for the HSE,the classical Lie point symmetry operators are obtained.Also,the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of onedimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed.Particularly,the Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained.Mainly,the conservation laws of the HSE are computed via three different methods including Boyer's generalization of Noether's theorem,first homotopy method and second homotopy method.     

New Exact Solutions and Conservation Laws to (3+1)-Dimensional Potential-YTSF Equation

Using the modified CK's direct method, we build the relationship between new solutions and old ones and find some new exact solutions to the (3+1)-dimensional potential-YTSF equation. Based on the invariant group theory, Lie point symmetry groups and Lie symmetries of the (3+1)-dimensional potential-YTSF equation are obtained. We also get conservation laws of the equation with the given Lie symmetry.     

Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg–de Vries equation

In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries(LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation, and second, we find the related exact solutions for the derived generators. Finally,according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field Kd V equation.     

Exhaustive Classification of the Invariant Solutions for a Specific Nonlinear Model Describing Near Planar and Marginally Long-Wave Unstable Interfaces for Phase Transition

Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dynamics can be modeled by an interface motion. This paper is devoted to exhaustive analysis of the invariant solutions for a modified Kuramoto-Sivashinsky equation in two spatial and one temporal dimensions is presented. This nonlinear partial differential equation asymptotically characterizes near planar interfaces, which are marginally long-wave unstable. For this purpose, by applying the classical symmetry method for this model the classical symmetry operators are attained.Moreover, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras, which yields the preliminary classification of group invariant solutions is constructed. Mainly, the Lie invariants corresponding to the infinitesimal symmetry generators as well as associated similarity reduced equations are also pointed out. Furthermore,the nonclassical symmetries of this nonlinear PDE are also comprehensively investigated.     

Symmetry properties,conservation laws and exact solutions of time-fractional irrigation equation

In this paper, we deal with the complete algebra of Lie point symmetries for the generalized model of an irrigation system of fractional order. By means of Lie symmetry method, the vector fields has been investigated which are utilized for obtaining the conservation laws of equation. In addition, through the sub-equation method, we construct some exact solutions for the considered equation by reducing the fractional partial differential equation to a ordinary fractional differential equation.     

Coupled Nonlinear Schrodinger Equation： Symmetries and Exact Solutions

The symmetries, symmetry reductions, and exact solutions of a coupled nonlinear Schrodinger （CNLS） equation derived from the governing system for atmospheric gravity waves are researched by means of classical Lie group approach in this paper. Calculation shows the CNLS equation is invariant under some Galilean transformations, scaling transformations, phase shifts, and space-time translations. Some ordinary differential equations are derived from the CNLS equation. Several exact solutions including envelope cnoidal waves, solitary waves and trigonometric function solutions for the CNLS equation are also obtained by making use of symmetries.     

Invariance Analysis and Variational Conservation Laws for the Wave Equation on Some Manifolds

In this paper we discuss symmetries of a nonlinear wave equation that arises as a consequence of some Riemannian metrics of signature −2. The objective of this study is to show how geometry can be responsible in giving rise to a nonlinear inhomogeneous wave equation rather than assuming nonlinearities in the wave equation from physical considerations. We find Lie point symmetries of the corresponding wave equations and give their solutions in two cases. Some interesting physical conclusions relating to conservation laws such as energy, linear and angular momenta are also determined.     

Lie point symmetries,conservation laws,and analytical solutions of a generalized time-fractional Sawada–Kotera equation

Under investigation in this work is the invariance properties of the time-fractional generalized Sawada–Kotera equation, which can describe motion of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice. With the help of the Lie symmetry analysis method of fractional differential equations, we strictly derive the vector fields and symmetry reductions of the equation. Furthermore, based on the power series theory, an effective method is presented to succinctly construct its analytical solutions. Finally, using the new conservation theorem, the conservation laws of the equation are well constructed with a detailed derivation.     

Conservation laws of the complex short pulse equation and coupled complex short pulse equations

In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated. The two complex nonlinear models are turned into multi-component real models by proper transformations. Lie symmetries are obtained via the classical Lie group method, and the results for the coupled complex short pulse equations contain the existing results as particular cases. Based on the linearizing operator and adjoint linearizing operator for the two real systems, adjoint symmetries can be obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair (SA) method. Relationships between the nonlinear self-adjointness method and the SA method are investigated.