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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 $u_{tt}$?

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$u_t=\partial u/\partial t$ is simply notation and means the time derivative to first, $u_t$, and second, $u_{tt}$ orders. –  Kyle Kanos Apr 1 at 13:18
    
Also, this recent question by BMS might be useful in seeing the difference between $u_t$ and $u_{tt}$. –  Kyle Kanos Apr 1 at 13:21

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