I have a system of partial differential equations $$\partial_t g(x,t)=-(-a \partial^2_x f(x,t)+V(x)f(x,t)), $$ $$\partial_t f(x,t)=(-a \partial^2_x g(x,t)+V(x)g(x,t)), $$ where $V(x)$ is known function and $a$ is a parameter.
I am wondering if there is a way to convert these equations into some equation for the ratio between the two functions $g(x,t)/f(x,t)$.
This PDE system is of first order in time. Often, such systems can be converted into a single PDE of higher order. To do so, let us differentiate the first equation with respect to $t$. Assuming the equality of mixed derivatives, $$ \partial_{tt} g = a \partial_{xxt} f - V \partial_t f = (a \partial_{xx} - V)\, \partial_t f \, . $$ Using the second equation $\partial_t f = -(a\partial_{xx} - V)\, g$, the following PDE is obtained: $$ \partial_{tt} g = -a^2\partial_{xxxx}g + 2aV\partial_{xx} g - V^2 g \, . $$ Similarly, one shows that $f$ satisfies the above PDE.