The solution to the non-linear convection equation

Question

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?

Answer 1

1. 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.

2. 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.

3. 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 $\xi$.

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

5. 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.

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

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