Travelling wave

The substitution $U(x,t)=U(x-ct)$ turns time derivatives into $-c{U}^{\prime }$ and spatial diffusion into $U^{\prime \prime }$, reducing a reaction-diffusion PDE to an ODE system.

The linear spreading speed is obtained from exponentially decaying modes at an unstable state's leading edge and the condition that their spatial exponents become repeated.

For diffusion coefficient $D$ and leading-edge growth rate $r>0$, a Fisher-KPP-type pulled front has minimum speed $2\sqrt{Dr}$.

A pulled front is selected by growth and diffusion in its small-amplitude leading edge rather than by nonlinear dynamics behind the front.