In my work I came across a second order partial differential equation with complex coefficients in three variables of the form (for which i am seeking an analytical solution):
$$\frac{\partial }{\partial t}P[x_{1}^{},x_{2}^{},t_{}^{}]=\sum_{i,j=1,2}\Bigg[\frac{\partial }{\partial x_{i}^{}}A_{ij}^{}\frac{\partial }{\partial x_{j}^{}}+x_{i}^{}B_{ij}^{}\frac{\partial }{\partial x_{j}^{}}+\frac{\partial }{\partial x_{i}^{}}B_{ij}^{T}x_{j}^{}+ x_{i}^{}C_{ij}^{}x_{j}^{}\Bigg]P[x_{1}^{},x_{2}^{},t_{}^{}]$$ with the boundary conditions $P[x_{1}^{},x_{2}^{},t_{}^{}]\Big|_{t=0}=P_{in}^{}[x_{1}^{},x_{2}^{}]$ and $P[x_{1}^{},x_{2}^{},t_{}^{}]_{x_{i}^{}=\pm\infty}=0$. Here matrices $A$, $B$ and $C$ are constant complex matrices.
I attempted (brute force approach) to solve it by a transformation to remove quadratic term (which leads to matrix a riccatti ordinary differential equation), followed by fourier transformation to convert to first order partial differential equation which can be solved by method of characteristics. Midway I realized (characteristics curves become complexified, legal issues of fourier transformability etc.,) that this brute force approach somehow cannot be justified (not sure, is this procedure alright?).
I would like to know both about the existence and uniqueness of solutions of this equation (if there are any existence and uniqueness theorems for second order partial differential equations with complex coefficients) and a concrete method to solve this particular partial differential equation with arbitrary constant complex matrices $A$, $B$ and $C$ (are there any references giving procedures to solve this kind of equations).
Thanks in advance.