Swift–Hohenberg equation
From HandWiki
The Swift–Hohenberg equation (named after Jack B. Swift and Pierre Hohenberg) is a partial differential equation noted for its pattern-forming behaviour. It takes the form
- [math]\displaystyle{ \frac{\partial u}{\partial t} = r u - (1+\nabla^2)^2u + N(u) }[/math]
where u = u(x, t) or u = u(x, y, t) is a scalar function defined on the line or the plane, r is a real bifurcation parameter, and N(u) is some smooth nonlinearity.
The equation is named after the authors of the paper,[1] where it was derived from the equations for thermal convection.
The webpage of Michael Cross[2] contains some numerical integrators which demonstrate the behaviour of several Swift–Hohenberg-like systems.
References
- ↑ J. Swift; P.C. Hohenberg (1977). "Hydrodynamic fluctuations at the convective instability". Phys. Rev. A 15 (1): 319–328. doi:10.1103/PhysRevA.15.319. Bibcode: 1977PhRvA..15..319S.
- ↑ Java applet demonstrations
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