Characteristic (Partial Differential Equation)

Paths in a 2-D plane used to transform Partial Differential Equations into systems of Ordinary Differential Equations. They were invented by Riemann. For an example of the use of characteristics, consider the equation

$$u_t-6uu_x=0.$$

Now let $u(s)=u(x(s),t(s))$. Since

$${du\over ds}={dx\over ds}u_x+{dt\over ds}u_t,$$

it follows that ${dt/ds}=1$, ${dx/ds}=-6u$, and ${du/ds}=0$. Integrating gives $t(s)=s$, $x(s)=-6su_0(x)$, and $u(s)=u_0(x)$, where the constants of integration are 0 and $u_0(x)=u(x,0)$.