两个非线性偏微分方程的分离变量解

Solutions of Two Nonlinear Partial Differential Equations in the Form of Separated Variables

三维旋度方程的一维模型研究中 ,引出的两个非线性偏微分方程 (PDE) ,分别被看做是Burgers方程和KdV方程的二维推广 ,它们都存在分离变量形式的精确解。这些解可分别借助线性热导方程和相应的线性KdV方程的解去构造。若给定分离变量形式的初值函数 ,则初值问题的精确解也是分离变量形式的。

Two nonlinear PDEs arising from one dimensional model for the three dimensional vorticity equation have exact solutions in the form of separated variables. Since these PDEs can be regarded as a generalization of Burgers equation and KdV equation to two dimensional cases, the construction of such solutions can be reduced to those of the linear heat conduction equation and linear KdV equation respectively. If the initial value function of an initial value problem is in the form of separated variables, then its exact solution is also in the form of separated variables.