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$$u_t+uu_x=0$$ (17)

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$$\left\{\begin{array}{l} \displaystyle \frac{d x}{d t}=u(t,x),\\ x(0)=x_0 \end{array}\right.$$ (18)

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$$\left\{\begin{array}{l} \phi'(t)=u(t,\phi(t)) \hspace{1zw}(t>0),\\ \phi(0)=x_0 \end{array}\right.$$ (19)

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$$\begin{eqnarray*} \frac{d v}{d t} & = & \frac{\partial u}{\partial t}(t,\phi(t)... ... \\ & = & \left. u_t+uu_x\right\vert _{x=\phi(t)} \\ & = & 0 \end{eqnarray*}$$

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$$\phi'(t)=u(t,\phi(t))=u(0,x_0)$$

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$$\phi(t)=\phi(0)+\int_0^tu(0,x_0)dt=\phi(0)+u(0,x_0)t = x_0 + u(0,x_0)t$$

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$$u_t+\alpha(t,x,u)u_x = \beta(t,x,u) \hspace{1zw} \left(\frac{\partial \alpha}{\partial u}\neq 0\right)$$

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$$x=f(x_0)t+x_0$$ (20)

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$$u=u(0,x_0)=f(x_0)$$ (21)

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