Pattern formation

Pattern formation is the development of spatial structure from an initially uniform state, often through loss of stability to perturbations with a preferred nonzero wavenumber. If a translation-invariant linearized equation has Fourier growth rate $\lambda (k)$, an unstable maximum near $k=k_c\ne 0$ initially amplifies spatial modes near that scale. Nonlinear interactions then determine whether the pattern saturates, drifts, oscillates or changes scale. The Swift–Hohenberg equation supplies a simple model with $\lambda (k)=r-(1-k^2{)}^2$, selecting $k\simeq 1$ near onset.

A planform is the horizontal spatial structure of a pattern, often described by the Fourier modes active at a bifurcation. For example, one pair of opposite wavevectors gives parallel convection rolls, while two equal-amplitude perpendicular pairs give squares. A planform includes the relative amplitudes and phases needed to distinguish spatial symmetry types.

A phase modulation changes the local position of a periodic pattern. Its local wavenumber is shifted by derivatives of the phase. In a method of multiple scales, a correction profile depending on the translated coordinate $\xi =x+\varphi (Y)$ must be differentiated with ${\mathcal{D}}_Y={\partial }_Y+{\varphi }_Y{\partial }_{\xi }$. Omitting this chain rule can change the nonlinear coefficient of a reduced amplitude equation even when its linear coefficients are unchanged.

Let $p={\varphi }_Y$, $r={\varphi }_{YY}$ and $w_2=C_1(\xi )r+C_2(\xi )p^2$. Two applications of the chain rule give ${\mathcal{D}}_Y^2{w}_2=C_1{\varphi }_{YYYY}+2(C_1^{\prime }+C_2)p{\varphi }_{YYY}+(C_1^{\prime }+2C_2)r^2+(C_1^{\prime \prime }+5C_2^{\prime })p^{2}r+C_2^{\prime \prime }{p}^4$. In a Fredholm solvability condition, the fourth term generally contributes to the coefficient of ${\varphi }_Y^2{\varphi }_{YY}$. It cannot be omitted merely because the fast-coordinate correction functions have been denoted by $C_j(x)$.

A zigzag instability is a long-wave transverse phase modulation of parallel rolls. In ${\varphi }_T=\lambda {\varphi }_{YY}-\gamma {\varphi }_{YYYY}-\alpha {\varphi }_Y^2{\varphi }_{YY}$, with $\gamma >0$, the straight roll loses transverse linear stability analysis when $\lambda <0$. Differentiating gives the conserved slope equation $p_T={\partial }_Y^2(\lambda p-\gamma p_{YY}-\alpha p^3/3)$. When $\lambda ,\alpha <0$, the nonzero preferred slopes satisfy $p^2=3\lambda /\alpha$.

At a transverse phase-diffusion threshold the coefficient of the second spatial derivative is of order ${\epsilon }^2$. Its product with two long-wave derivatives therefore has order ${\epsilon }^4$, the same order as the stabilizing fourth derivative. This balance determines the slow time $T={\epsilon }^{4}t$ and retains both terms in the leading phase modulation equation.

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 $A_T=\mu A+3\hat{s}A|A{|}^2-10A|A{|}^4+4A_{XX}$.