Solving a Higher-Order PDE: Traveling Wave & Phase Portrait

[squenshl]

Homework Statement

Consider the PDE u_t + 6u³u_x + u_xxx = 0
which may be thought of as a higher-order variant of the KdV.
a) Assume a traveling wave u = f(x-ct) and derive the 3rd-order ODE for that solution.
b) Reduce the order of this ODE and obtain the expression for the polynomial g(f), where g(f) = (f')²/2
c) Sketch the (f,f') phase portrait for real solutions, assuming that g(f) has the maximum number of real roots.

Homework Equations

The Attempt at a Solution

a) I let u = f(x-ct) and got -cf'(x-ct) + 6f(x-ct)³f'(x-ct) + f'''(x-ct) = 0, is this the 3rd-order ODE required.
b) I got (f')²/2 = g(f) where g(f) = -f(x-ct)⁵/4 + cf(x-ct)²/6 + A₁f(x-ct) + A₂
c) I know that this has 5 solutions (5th-order polynomial), do I write it in the form (f')²/2 = g(f) = 1/4*(f-F₁)(f-F₂)(f-F₃)(f-F₄)(F₅-f) and if so how does this look on phase portrait.

 

[squenshl]

I just need to do c). Any ideas?