The solution to the non-linear convection equation The non-linear convection equation
$$u_{t} +uu_{x}=0$$ admits implicit solutions of the form $$u=f(x-ut).$$
How does one interpret this solution intuitively? Is there an example of a solution of this equation that can  be written in an explicit form? How does an initial function evolve over time? 
 A: *

*This quasi-linear 1st-order PDE
$$\left(\frac{\partial u}{\partial t}\right)_{\!x} + u\left(\frac{\partial u}{\partial x}\right)_{\!t}~=~0\tag{1}$$
is the inviscid Burgers' equation. 

*It can be solved via the method of characteristics. The ODE IVP
$$ \frac{dx}{dt}~=~u, \qquad x(t\!=\!0)~=~\xi, \tag{2}$$
(where $u$ is treated as an external parameter and $\xi$ is an initial value) has integral curves
$$ x(t) ~=~ u t + \xi. \tag{3}$$
The initial value $\xi$ and the external parameter $u$ label the integral curves (3) in an $(t,x)$-diagram.

*A solution $u(x,t)$ to the PDE is constant along above integral curves (3):
$$ \frac{du(x(t),t)}{dt}~=~\left(\frac{\partial u}{\partial t}\right)_{\!x} + \left(\frac{\partial u}{\partial x}\right)_{\!t}\frac{dx}{dt}~\stackrel{(2)}{=}~ \left(\frac{\partial u}{\partial t}\right)_{\!x} + u\left(\frac{\partial u}{\partial x}\right)_{\!t}~\stackrel{(1)}{=}~0.\tag{4}$$
This leads to that the solution $u=f(\xi)$ should be a function of $\xi$. 

*The function 
$$x\quad \mapsto\quad f(x)~=~u(x,t\!=\!0)\tag{5}$$ 
is the initial profile. 

*The implicit solution 
$$u(x,t)~=~f(x-t u(x,t))~=~f(x-t f(x-t u(x,t)))~=~\ldots\tag{6}$$
is a fixed-point equation.

*Interestingly, the non-linear PDE can develop shock-wave singular solution $u$ from a smooth regular initial profile $f$.

*Example. The Wikipedia page lists the full solution 
$$u(x,t)~=~\frac{ax+b}{at+1}\tag{7}$$ for the initial profile 
$$f(x)~=~ax+b.\tag{8}$$
