Help me solve question

Consider the following equation with constant α.
ut = (f(ux))x − αuxxxx, f(v) = v2 − v
A. Linearize this equation around u = 0 and find the principal mode solution of the form eωt ikx. For which values of α are there unstable modes, i.e., modes with ω = 0 for real k? For these values, find the maximally unstable mode, i.e., the value of k with the largest positive value of ω.
B. Consider the steady solution of the (fully nonlinear) problem. Show that the resulting equation can be written as a second order autonomous ODE for v = ux and draw the corresponding phase plane.