I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose as part of some recent work. I have a relatively simple (I think) 2D system of PDEs of the form:

$\frac{\partial u_1}{\partial t}+\left(f_1\left(t\right)+B_1\right)\frac{\partial u_1}{\partial x}=-u_1+u_2+C_1f_1\left(t\right)e^{Ax}\\ \frac{\partial u_2}{\partial t}-\left(f_2\left(t\right)+B_2\right)\frac{\partial u_2}{\partial x}=Du_1-Du_2+C_2f_2\left(t\right)e^{Ax}$

Where all the capital letters are constants and the $f$ functions are such that you always have real eigenvalues. I've been trying to follow the method using characteristic invariants outlined in this document and came up with a few questions.

First off, am I wrong to suspect that there may be an explicit solution to this system? Integral form or otherwise. Second, I came across some notation the authors use first on page 3 that I don't fully understand. It is:

$L\left(h\right)|_{[S]}=0$

Where $L$ is an operator, $h$ is a solution, and where $[S]$ means "the system and its differential consequences with respect to $x$." As I read it, it seems to be a restriction where only the $x$ differentiation is considered, but I'm really unsure as to what is meant by it. They give an example a few paragraphs below and develop an operator of the form $L_2=D_t+(u+c)D_x$ and solve the above equation using that operator, but the details are omitted and I haven't been able to fill them in myself.

Any help you could offer would be great. Even if there's no hope of solving the system analytically, it would be helpful to get a better understanding of that notation for the future. Thanks in advance.

I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose as part of some recent work. I have a relatively simple (I think) 2D system of PDEs of the form: $$ \begin{cases} \dfrac{\partial u_1}{\partial t}+\big(f_1(t)+B_1\big)\dfrac{\partial u_1}{\partial x}=-u_1+u_2+C_1 f_1(t)e^{Ax}\\ \\ \dfrac{\partial u_2}{\partial t}-\big(f_2(t)+B_2\big)\dfrac{\partial u_2}{\partial x}=Du_1-Du_2+C_2f_2(t)e^{Ax} \end{cases} $$ Where all the capital letters are constants and the $f$ functions are such that you always have real eigenvalues. I've been trying to follow the method using characteristic invariants outlined in this document and came up with a few questions.

First off, am I wrong to suspect that there may be an explicit solution to this system? Integral form or otherwise. Second, I came across some notation the authors use first on page 3 that I don't fully understand. It is: $$ L\left(h\right)|_{[S]}=0 $$ Where $L$ is an operator, $h$ is a solution, and where $[S]$ means "the system and its differential consequences with respect to $x$." As I read it, it seems to be a restriction where only the $x$ differentiation is considered, but I'm really unsure as to what is meant by it. They give an example a few paragraphs below and develop an operator of the form $$ L_2=D_t+(u+c)D_x $$ and solve the above equation using that operator, but the details are omitted and I haven't been able to fill them in myself.

Any help you could offer would be great. Even if there's no hope of solving the system analytically, it would be helpful to get a better understanding of that notation for the future. Thanks in advance.