Using Darboux's to solve 2D system of first order linear PDEs with variable coefficients

Question

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.

Answer 1

This isn't a solution, but it's too long for a comment. Before you try to apply Darboux' Method, you might want to clean up your system a bit.

First, notice that this is an inhomogeneous linear system for the pair of functions $u_1$ and $u_2$. If you look for a particular solution in the form $$ u_1(x,t) = h_1(t)\mathrm{e}^{Ax}\quad\text{and}\quad u_2(x,t) = h_2(t)\mathrm{e}^{Ax}, $$ you'll get an inhomogeneous linear first order system of ODEs for $h_1$ and $h_2$, and it clearly has global solutions (though there's no 'explicit' general solution). Subtracting a particular solution of this ODE system from the general solution will give you a homogeneous linear system, which effectively, sets $C_1=C_2=0$.

With $C_1=C_2=0$, there's no reason not to incorporate the constants $B_1$ and $B_2$ into $f_1(t)$ and $f_2(t)$, so, without loss of generality, you can assume that $B_1$ and $B_2$ are zero.

Now, you will run into trouble at times $T$ where $f_1(t)+f_2(t) = 0$. These are times where the characteristics are no longer distinct. (Of course, if $f_1+f_2$ vanishes identically, then you actually have an ODE system in the right coordinates, so that's an easy case.) Let's assume that $f_1(t)+f_2(t)$ is nonvanishing. Then the system is hyperbolic. Moreover, in this case, we can establish new independent coordinates $$ y_1 = x - F_1(t)\quad\text{and}\quad y_2 = x + F_2(t) $$ where $F'_i = f_i$, and, letting $h$ be the function of one variable that satisfies $f_1 + f_2 = h(F_1+F_2)$, we see that the system reduces to the linear hyperbolic system $$ \frac{\partial u_1}{\partial y_2} = \frac{u_1{-}u_2}{h(y_2{-}y_1)} \quad\text{and}\quad \frac{\partial u_2}{\partial y_1} = D\,\frac{u_1{-}u_2}{h(y_2{-}y_1)}. $$

Now everything depends on the single constant $D$ and the single function $h$. One can now relatively easily check for Darboux integrability at the first few stages. I would expect that, for the generic $D$ and $h$, the above system is not integrable by Darboux' Method, but there might be special cases of $(D,h)$ where Darboux' Method succeeds at some level $k\ge0$.