Front (physics)

The most simple example of front solution connecting a homogeneous stable state with a homogeneous unstable state can be shown in the one-dimensional Fisher–Kolmogorov equation:

${\displaystyle N_{t}=DN_{xx}+rN(N_{0}-N)}$

that describes a simple model for the density ${\displaystyle N(x,t)}$ of population. This equation has two steady states, ${\displaystyle N=0}$, and ${\displaystyle N=N_{0}}$. This solution corresponds to extinction and saturation of population. Observe that this model is spatially-extended, because it includes a diffusion term given by the second derivative. The state ${\displaystyle N\equiv N_{0}}$ is stable as a simple linear analysis can show and the state ${\displaystyle N=0}$ is unstable. There exist a family of front solutions connecting ${\displaystyle N=N_{0}}$ with ${\displaystyle N=0}$, and such solution are propagative. Particularly, there exist one solution of the form ${\displaystyle N(t,x)=N(x-vt)}$, with ${\displaystyle v}$ is a velocity that only depends on ${\displaystyle D}$ and ${\displaystyle r}$[5]

Front solution connecting two steady states in a generic spatially extended system.

Propagative front profile