wave front solutions of the reaction-advection equation

Question

I do not understand how to approach the follwoing question:

What are the wave front solutions of the reaction-advection equation $u_t + u_x = u(1 − u)$?

Answer 1

I suppose that by a wave front you mean a solution of the form $u(x,t)=\phi(x-c\,t)$ for some function $\phi$ (the shape of the wave front) and constant $c$ (the speed of the wave front.) Substituting into the equation we get the ordinary differential equation $$ (1-c)\phi'=\phi(1-\phi). $$ It can be solved by separation of variables.