平面上抛物型二阶算子的微分变换
Differential transformations of parabolic second-order operators in the plane
摘要：
本文将Darboux关于双曲型方程微分替换变换的经典结果推广到抛物型方程$L u = \big(D^2_{x} + a(x,y) D_x + b(x,y) D_y + c(x,y)\big)u=0$的情形。我们证明了一个一般定理，提供了一种方法来确定上述抛物方程的转换。结果表明，高阶变换算子$M$总是可以表示为连续定义一系列变换的一阶算子的组合。逆变换的存在意味着对初始算子系数的一些微分约束。我们表明，这些关系可以暗示著名的可积方程，特别是Boussinesq方程。
英文摘要：
Here, Darboux's classical results about transformations with differential substitutions for hyperbolic equations are extended to the case of parabolic equations of the form $L u = \big(D^2_{x} + a(x,y) D_x + b(x,y) D_y + c(x,y)\big)u=0$. We prove a general Theorem that provides a way to determine transformations for parabolic equations shown above. It turnes out that transforming operators $M$ of some higher order can be always represented as a composition of some first-order operators that consecutively define a series of transformations. Existence of inverse transformations implies some differential constrains on the coefficients of the initial operator. We show that these relations can imply famous integrable equations, in particular, the Boussinesq equation.
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