# 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?

## 1 Answer

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 $$f(x)~=~ax+b.\tag{8}$$