I have an equation of the form
$$ \left(\frac{\partial }{\partial t} + a \frac{\partial }{\partial x} \right) \phi + \eta \left(\frac{\partial }{\partial t} + c_1 \frac{\partial }{\partial x} \right) \left(\frac{\partial }{\partial t} + c_2 \frac{\partial }{\partial x} \right) \phi = 0, $$
where $\eta$ is a positive constant and $c_1>c_2$.
When $\eta \to 0$ we have the general solution using the method of characteristics
$$ \phi= \phi_0 (x - a t), $$
and when $\eta \to \infty$ we have the general solution using the method of characteristics
$$ \phi= \phi_1 (x - c_1 t) + \phi_2 (x - c_2 t). $$
Is it possible to find a general solution for $\phi$ without having to take a limit of $\eta$ (possibly using the method of characteristics or a Fourier transform)?