TITLE: Simple PDE classification question
QUESTION [5 upvotes]: Benjamin-Bona-Mahony equation:
$$\displaystyle u_t+u_x+uu_x-u_{xxt}=0$$
both the paper I was reading and wikipedia claimed that it is nonlinear. It has been some time since I studied classification so I wasn't sure it is nonlinear because of $uu_x$ or $u_{xxt}$ or both.
So I looked it up and found that
"... semilinear equations are ones in which the coeﬃcients of
the terms involving the highest-order derivatives of u depend only on x, not on u or its
derivatives."
And claims that $$\displaystyle u_t+u_{xxx}+u u_{x}=0$$ 
is semiliner.
So it seems to me that if BBM is indeed nonlinear, it is not because of $uu_{x}$, but because of $u_{xxt}$, but that feels unnatural for me.
Please shed some light on me.

REPLY [18 votes]: Consider first order PDE depending on two independent variables. It is

linear, if it has the form
$$
a(x,y)\partial_x u+b(x,y)\partial_yu=c(x,y)u+f(x,y),
$$
Example:
$$
\partial_xu+\partial_yu=0.
$$
semi-linear if it has the form
$$
a(x,y)\partial_x u+b(x,y)\partial_yu=c(x,y,u),
$$
Example:
$$
\partial_xu+\partial_yu=u^2
$$
quasi-linear if it has the form
$$
a(x,y,u)\partial_x u+b(x,y,u)\partial_yu=c(x,y,u),
$$
Example:
$$
\partial_xu+u\partial_yu=0
$$
fully non-linear if it has the form
$$
F(x,y,u,\partial_xu,\partial_yu)=0.
$$
Example:
$$
(\partial_xu)^2+(\partial_yu)^2=V(x,y).
$$

Note that semi-linear, quasi-linear, and fully nonlinear equations are nonlinear.
Can you generalize this classification to equations of any order?