Front (physics)

In physics, a front^[1]^[2] is a solution of an spatially extended system connecting two steady states. From dynamical systems point of view, a front correspond to a heteroclinic orbit of the system in the co-mobile frame (or proper frame).

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

Propagative front profile

Fronts connecting stable - unstable homogeneous states

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:

u_{t}=Du_{xx}+ku(1-u)

that describes a simple model for the density $u(x,t)$ of population. This equation has two steady states, $u\equiv 0$ , and $u\equiv 1$ . 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 $u\equiv 0$ is stable as a simple linear analysis can show and the state $u\equiv 1$ is unstable. There exist a family of front solutions connecting $u=1$ with $u=0$ , and such solution are propagative. Particularly, there exist one solution of the form $u(t,x)=u(x-vt)$ , where $v$ constants, depending only on the coefficient of diffusion and growth constant k.

References

↑ Pismen, L. M. (2006). Patterns and interfaces in dissipative dynamics. Berlin: Springer. ISBN 978-3-540-30430-2.
↑ Horsthemke, Vicenç Mendéz, Sergei Fedotov, Werner (2010). Reaction-transport systems : mesoscopic foundations, fronts, and spatial instabilities. Heidelberg: Springer. ISBN 978-3642114427.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Pismen, L. M. (2006). Patterns and interfaces in dissipative dynamics. Berlin: Springer. ISBN 978-3-540-30430-2.

[2] Horsthemke, Vicenç Mendéz, Sergei Fedotov, Werner (2010). Reaction-transport systems : mesoscopic foundations, fronts, and spatial instabilities. Heidelberg: Springer. ISBN 978-3642114427.