Non linear pde need to change it to an ode

[maggie56]

Homework Statement

my non linear pde is
du/dt = d/dx [3u² - d²u/dx² ]
The question says to let u(x,t) = f(x-ct)
Where the function f tends to 0, f' tends to 0 and f'' tends to 0 but the (x-ct) tends to positive or negative infinity.

Homework Equations

i thought the solution was to find du/dt du/dx d²u/dt² and d²u/dx²
Which gave me in order as above, -cf' , f' , c²f'' , f''

The Attempt at a Solution

But when i substitute these into the pde i get -cf' = d/dx[3f² - f"] which I am not sure actually helps

Could someone please tell me if i am using the correct method or if i have done it completely wrong

Thank you