Decoupling a set of four first-order PDEs

Question

I have the following four PDEs:

$$ r_x = -\frac{i}{2} r \left(2\lambda ^2-1\right) + e^{+i x} s \lambda\, ,\\ s_x = +\frac{i}{2} s \left(2\lambda ^2-1\right) - e^{-i x} r \lambda \, , \\ r_t = +e^{+i x} s - i r \lambda \, , \\ s_t = -e^{-i x} r + i s \lambda \, ,\\ $$

How should I proceed with decoupling them or solving them in any other way?

$\lambda$ is generally complex and independent of $x$ and $t$, but I could settle for solving them for purely imaginary $\lambda$. In that case, I know the solution has the form $2i\exp(-ix/2)\sin(A+iB)$ where A and B are some real functions of $(x,t,\lambda)$ (I can provide their exact form if needed).

Answer 1

We observe that four equations are too much to be solved for two unknown functions only $r(x,t)$ and $s(x,t)$ .

This is consistent insofar $r_{ts}=r_{st}$ and $s_{ts}=s_{st}$. One can directly verify these relationships, or alternatively solve the system without using the four given equations and test the consistency at the end in bringing back the result into the unused equations.