characteristic curve

A curve along which a first-order PDE reduces to an ordinary differential equation for u.

For a first-order PDE $a(x,y)u_x+b(x,y)u_y=c(x,y,u)$ , a characteristic curve is a curve $(x(t),y(t))$ solving $x'(t)=a(x,y)$ , $y'(t)=b(x,y)$ ; along it, $u$ satisfies the single ODE $u'(t)=c(x(t),y(t),u(t))$ , so the PDE becomes an ordinary differential equation one characteristic at a time.