论文标题
显着（1+2）的扩展对称分析 - 维fokker-planck方程
Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation
论文作者
论文摘要
我们对具有三个独立变量的超帕代谢代谢fokker-Planck方程进行了扩展对称分析，这也称为Kolmogorov方程，并通过其显着的对称属性在此类Fokker-Planck方程的类别中列出。特别是，其必需的谎言不变性代数是八维，这是上述类中的最大维度。我们使用直接方法来计算Fokker-Planck方程的完整点对称性伪集团，分析其结构并挑出其必需子组。在列出了该方程的基本和最大谎言代数的不相等的一维次代代代代数之后，我们对其谎言减少，进行了奇特的广义减少，并将后者的降低与生成溶液与地面运营商的迭代作用相关联。结果，我们构建了Fokker-Planck方程的精确溶液的广泛家族，特别是通过（1+1） - 维线性线性热方程的任意数量的任意溶液进行参数。我们还建立了fokker-Planck方程与（1+2）维克拉默斯方程的点相似性，其基本谎言不变性代数是八维的，这使我们能够以一种简单的方式找到这些Kramers方程的精确解决方案。
We carry out the extended symmetry analysis of an ultraparabolic Fokker-Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker-Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker-Planck equation using the direct method, analyze its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalized reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker-Planck equation, in particular, those parameterized by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker-Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.
