View source for Swift-Hohenberg equation

==Motivation== 

The motivation for the study which led to the SH equation was the analogy between bifurcations in the hydrodynamic behavior of fluids and the associated pde’s on the one hand, and continuous phase transitions in thermodynamic systems on the other hand. 

The specific physical system focused on was Rayleigh-B&eacute;nard (RB) convection which had attracted renewed interest among physicists in the early 1970s (Ahlers, 1974).

Classical bifurcation theory is analogous to the mean-field or Landau theory of phase transitions which neglects fluctuation effects.  The two theories give the same values for the exponents which describe the scaling of the variables of the theory near the bifurcation or transition point. Classical theory and experiment reveal that the amplitude of the fluid velocity field increases continuously from zero as the square root of the difference of the control parameter and its critical value near the onset of RB convection, and the spontaneous magnetization increases as the square root of the difference of the  critical temperature and the temperature, in a Heisenberg ferromagnet according to mean-field theory.

It was natural at that time (1974) to ask whether such fluctuations, whose dramatic effects on continuous thermodynamic transitions had been so successfully elucidated by the scaling and renormalization theories in the period 1965-1975, might also modify the behavior of fluids near hydrodynamic bifurcation points.