Consider the PDE ut + 6u3ux + uxxx = 0
which may be thought of as a higher-order variant of the KdV.
a) Assume a travelling 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.
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.