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## Homework Statement

Consider the PDE u

_{t}+ 6u

^{3}u

_{x}+ u

_{xxx}= 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.

## Homework Equations

## The Attempt at a Solution

a) I let u = f(x-ct) and got -cf'(x-ct) + 6f(x-ct)

^{3}f'(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 + A

_{1}f(x-ct) + A

_{2}

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-F

_{1})(f-F

_{2})(f-F

_{3})(f-F

_{4})(F

_{5}-f) and if so how does this look on phase portrait.