Perturbative solution of a PDE I'm in trouble with the following $PDE$ $$\partial_t\rho(x,t)=(1-2\rho(x,t))\partial_x\rho(x,t)$$ I'm finding solutions of the previous equation given the following initial condition: $$\rho(x,0)=\rho_0+\epsilon\rho_1(x)$$ where $\rho_0$ is a constant and $\rho_1(x)=sin(\omega x)$ Can I use a perturbative approach if $\epsilon$ is small enough? If the answer is 'yes', how much $\epsilon$ has to be small in order to consider the perturbative approach good, even if I understand the word 'good' could be misleading? Thanks.

In fact this belongs to a PDE of the form <

The solution of this PDE question is $\begin{cases}\rho=\rho_0+\epsilon\sin\omega u\\\x=u+(1-2(\rho_0+\epsilon\sin\omega u))t\end{cases}$