= Cubic-quintic Swift–Hohenberg amplitude reduction
{title2=$A_T=\mu A+3\hat sA|A|^2-10A|A|^4+4A_{XX}$}
In the mildly subcritical <Swift–Hohenberg equation>, set $r=\varepsilon^4\mu$, $s=\varepsilon^2\hat s$ and $w=\varepsilon[A(X,T)e^{ix}+\overline A e^{-ix}]+\cdots$. The nonlinear and growth terms first balance at order $\varepsilon^5$, so $T=\varepsilon^4t$. Quadratic detuning of the critical Fourier growth rate gives $X=\varepsilon^2x$. At order $\varepsilon^5$, 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 $A_T=\mu A+3\hat sA|A|^2-10A|A|^4+4A_{XX}$.
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