偏微分方程论文：偏微分方程孤立子解Lie变换群

偏微分方程论文：偏微分方程孤立子解 Lie变换群

【中文摘要】本文取得的主要结果属于理论性的,可概括如下:首先利用推广的Tanh-函数法以及在此基础上的拓展和形变映射法,获得了BBM方程的许多显式精确行波解,包括孤子解、复线孤子解、周期波解、Jacobi椭圆函数解、维尔斯特拉斯椭圆函数解等。其次介绍如何利用Lie变换群作用下偏微分方程的不变性来构造它的解。与常微分方程的情形相似,我们将看到,确定一个给定PDE所拥有的Lie点变换群的无穷小生成元,其算法可由它的不变性无穷小准则直接导出。利用Lie对称群的不变曲面可得到相似解,这样的解是通过求解约化方程得到的。约化方程所含未知变量个数比原方程少。本节就是用古典无穷小算法导出了由轴对称波方程的任意元和无穷小生成子的系数构成的超定线性偏微分方程组,即确定方程DE。其次借助符号计算机软件maple解方程组,求出了轴对称波方程的一些无穷小生成元,然后根据Lie第一基本定理求出了相对应的单参数Lie变换群.最后将所求得的无穷小生成元代入不变曲面条件,分别利用不变形式法和直接代入法求出轴对称波方程的群不变解。最后讨论如何利用Lie点变换群作用下的不变性求解PDEs的边值问题。如果PDE所拥有的单参数Lie点对称群同时也使边值问题的边界条件和领域不变,那么此边值问题的解也是不变解。因此,边值问题也可被构造性地约化为含更少的自变量的PDEs的边值问题。对于线性PDE,限制条件可放宽,不必要求边界条件不变。对应于同一特征函数展开的不变解

进行叠加。可得边值问题的解,其中特征值是利用一个齐次线性PDE 在其自变量的标度下的不变性得到的。另外,也将讨论多参数Lie点变换群作用下边值问题的不变性。我们利用上面给出的方法求出了Green函数的边值问题的不变解。

【英文摘要】First tanh-function method is extended then used to solve BBM equation. we also used deformation mapping method to obtain solutions of BBM equation. With both methods we can obtain abundant explicit and exact traving wave solutions. Which coation Soliton solutions, Plural line soliton solutions, periodic wave solutions, Jacobi elliptic fuction solutions,Weierstrass elliptic function solutions and other exact solutions.Second we apply infinitesimal transformations to the construction of solutions of partial differential equations. As for ODE’s we will show that the infinitesimal criterion for invariance of PDE’s leads directly to an algorithm to determine infinitesimal generators X admitted by given PDE’s . Invariant surfaces of the corresponding Lie group of point transformations lead to similarity solutions. These solutions are obtained by solving PDE’s with fewer independent variables than the given PDE’s. Now we obtain the set of determining equations is an overdermined system of PDE’s which is composed of the arbitrary

element of axisymmetric wave equation and the coefficient of infinitesimal generators, that derived by classical

infinitesimal Lie method. Second we give some infinitesimal generators of axisymmetric wave equation with the help of

symbols computer sorftware, after we find out the PDE’S

one-parameter Lie group of transformations by first

fundamental theorem of Lie. Last take the infinitesimal

generators that we find out into invariant surface condition

then we can get group invariant solutions of axisymmetric wave

equation by use invariant form method or direct

st we discuss how one can use infinitesimal transformations to solve boundary value problems for PDE’s .If

a one-parameter Lie group of transformation admitted by a PDE

leaves the domain and boundary conditions of a BVP invariant ,

then the solution of the BVP is an invariant solution, and hence

the given BVP is reduced to a BVP with one less independent

variable .we also consider the invariant of BVP’s under

multi-parameter Lie groups of transformations. We now apply the

given method to solve the boundary value probolems’solutions

of Green function.

【关键词】偏微分方程孤立子解 Lie变换群

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