Swift–Hohenberg equation ($w_t=[r-(1+{\partial }_x^2{)}^2]w+s{w}^3-w^5$)

The Swift–Hohenberg equation is a pattern-forming dissipative PDE with linear operator $r-(q_0^2+{\partial }_x^2{)}^2$. A spatial Fourier mode of wavenumber $k$ has growth rate $r-(q_0^2-k^2{)}^2$, selecting modes near $k=q_0$. A cubic-quintic version $w_t=[r-(1+{\partial }_x^2{)}^2]w+s{w}^3-w^5$ has destabilizing cubic feedback for $s>0$ and quintic saturation. Its weakly nonlinear envelope obeys a quintic real Ginzburg-Landau equation.

In the mildly subcritical Swift–Hohenberg equation, set $r={\epsilon }^4\mu$, $s={\epsilon }^2\hat{s}$ and $w=\epsilon [A(X,T)e^{ix}+\overline{A}{e}^{-ix}]+\cdots$. The nonlinear and growth terms first balance at order ${\epsilon }^5$, so $T={\epsilon }^{4}t$. Quadratic detuning of the critical Fourier growth rate gives $X={\epsilon }^{2}x$. At order ${\epsilon }^5$, projection onto the kernel of the self-adjoint fast operator $(1+{\partial }_x^2{)}^2$ requires the resonant $e^{\pm ix}$ forcing to vanish. The cubic and quintic resonances have coefficients $3A|A{|}^2$ and $10A|A{|}^4$, while the operator expansion gives $4A_{XX}$. Thus $A_T=\mu A+3\hat{s}A|A{|}^2-10A|A{|}^4+4A_{XX}$.