If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where $x\in[-L,L]$ and $t\in (0,T)$, $(h_0(t),h_1(t))$ are solutions to the system ${\bf h}'(t)={\bf M}{\bf h}(t)$. I have the explicit expressions for $h_0(t)$ and $h_1(t)$.
Is there a way of decoupling these PDEs? I think there is a way which could take advantage of the simpler $v$ equation.