I'm new to PDEs and need help linearizing this system around some equilibrium $(w_0, u_0)$:
$$\partial_t w = \partial_x u + w u$$ $${\partial_t}u = w({\partial_x}w) + b$$
where $u = u(x,t)$, $w = w(x,t)$, $b=b(x)$. I'm having a hard time dealing with the $w({\partial_x}w)$ term particularly. Also, if we are ignoring some partial derivative terms while doing the linearization, why are we ignoring them?
If $w_0$ and $u_0$ are an equilibrium solution to the system of PDEs, then they satisfy $$ \begin{cases} \partial_xu_0+w_0u_0=\partial_tw_0=0, \\ w_0\partial_xw_0+b=\partial_tu_0=0. \end{cases} \tag{1} $$ To linearize the original system around $(w_0,u_0)$, substitute $w$ and $u$ with $w_0+w_1$ and $u_0+u_1$, respectively, then use $(1)$ to eliminate terms that don't depend on $w_1$ and $u_1$, and discard terms that are second order in $w_1$, $u_1$ or their derivatives. The result is $$ \begin{cases} \partial_tw_1=\partial_xu_1+w_0u_1+w_1u_0, \\ \partial_tu_1=w_0\partial_xw_1+w_1\partial_xw_0. \end{cases} \tag{2} $$