Like we know, the standard form of KdV equation is
$$u_{t}-6uu_{x}+u_{xxx}=0,\tag{1}$$
where this equation describes a solitary wave propagation and $u=u(x,t)$.
On the other hand, we know the classical wave equation
$$\frac{{\partial}^{2}u}{{\partial}t^{2}}-\frac{{\partial}^{2}u}{{\partial}x^{2}}=0.\tag{2}$$
My question is: what is the physical difference exhibited by the terms $u_{t}$ and $u_{tt}$ in both equations? I mean, I know (1) is nonlinear what is the physical difference?
What does mean, physically, $u_{t}$ and $u_{tt}$?
