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What is Extended Fisher-Kolmogorov (EFK)?

There are many references to it, but I failed to find what is actually is. Is it some kind of non-linear differential equation? Does it describe physical systems?

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Short answer would be:

It is a nonlinear PDE, usually stated as

$u_t+\gamma u_{xxxx}-u_{xx}-u+u^3=0 $

as introduced in Dee & van Sarloos' paper. Most authors like to emphasize its application in modeling of phase transitions at critical points.