General solution to a n-dimensional partial differential equation

Question

$$ \begin{split} \frac{\partial}{\partial t}P(x, t)& =\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,\bigg(c_i\frac{\partial P}{\partial x_i} - c_j\frac{\partial P}{\partial x_j}\bigg)\\ &\qquad+\frac 12\sum\limits_{i<j}^{n}a_{i,j}\bigg(\frac{x_i-x_j}{1-c_i-c_j}\bigg)^2\bigg(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}\bigg)\\ &\qquad\qquad -\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} $$

I have been trying to solve the above PDE, with different methods such as Fourier Transform and method of separation of variables, but was not able to find a solution. What is the best way to find a general solution to the above PDE?

Answer 1

Denote $x_{ij} = x_i-x_j$, $\ n_{ij} = \frac{c_i+c_j}{1-c_i-c_j}$, and operator $$ D_{ij} = \frac{x_i-x_j}{1-c_i-c_j}(c_i\partial_i-c_j\partial_j). $$ We can write the pde as $$ P_t = \sum_{i<j}a_{i,j}\big( D_{ij}+\tfrac{1}{2}D_{ij}^2 \big)P. $$ It might remind one of the classical ``equidimensional" ode operators $x^2y''+axy'+by$ that preserve powers of $x$. If so, we can look for simple solutions of the form $e^{kt}x_{ij}^q$. We find that the functions $$ e^{a_{i,j}n_{ij}(q+\tfrac{1}{2}q^2)t} x_{ij}^q $$ solve the pde, so all linear combinations of these do also. Presumably you want $q$ to be positive integers.

I can't address to what extent these solutions are general. It would be helpful to specify something about initial and boundary conditions to do that.