Eliminate all parameters from the differential equation $u_t-Au_x-Bu^3+Cu_{xx}=0$.

Question

The question is to scale the equation $$u_t-Au_x-Bu^3+Cu_{xx}=0$$ to eliminate all parameters.

where $D>0$ and $A,B$ are nonzero.

I tried to substitute $U=U(x/L,t/t_0)$ into the equation and tried to find the appropriate value of $L,t_0$. I get $U_{\tau}t_0^{-1}=AU_{\xi}L^{-1}-bU^3+DL^{-2}U_{\xi\xi}$.

Then I tried but don't know how find the appropriate $t_0$ and $L$ to eliminate all parameters. Could anyone kindly help? Thanks!

Answer 1

Scale: $u = \frac{A}{\sqrt{BC}} U$, $t = \frac{C}{A^2} \tau, x = \frac{C}{A} \xi$.

To find this notice that you can adjust the scales of $u, t, x$, make such general scaling and solve for the conditions needed to factor the same prefactor out of all the equation. With the substitution above this prefactor should be

$$ \frac{A^3}{C\sqrt{BC}} = \Lambda $$ Notice that you will also need $B>0$ to this end. Your equation becomes:

$$ \Lambda (U_{\tau} - U_{\xi} - U^3 + U_{\xi \xi}) = 0 $$ dividing by $\Lambda$ also requires $A \neq 0$, and gives the equation you are looking for.