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For a stationary finite-wavenumber instability, the relevant Landau–Ginzburg model is the real Ginzburg–Landau equation: its coefficients are real although its pattern amplitude is complex. The complex amplitude records both the strength and the position of the underlying pattern. Assume a simple critical pair of Fourier modes $\pm k_c$, a quadratic nondegenerate maximum of the linear growth rate, and no additional conserved or marginal mode. The last assumption is essential: the conserved field in the previous problem requires an additional evolution equation.

To derive the model, write a real evolution equation schematically as $u_t=L(r)u+N_2(u,u)+N_3(u,u,u)+\cdots$, with symmetric multilinear nonlinearities. Put $r={\epsilon }^2{r}_2$, $X=\epsilon x$, $T={\epsilon }^{2}t$ and use the method of multiple scales expansionHere $L(k_c;0)v=0$. At the next order the noncritical mean and second harmonic are slaved to the critical mode:where $L(0;0)w_0=-2N_2(v,\overline{v})$ and $L(2k_c;0)w_2=-N_2(v,v)$. These inverses exist under the no-other-critical-mode hypothesis. Let $v^{\ast }$ be an adjoint null vector normalized by $⟨v^{\ast },v⟩=1$. At order ${\epsilon }^3$, the Fredholm solvability condition requires the resonant forcing to be orthogonal to $v^{\ast }$. The cubic resonant contribution isThis shows why the cubic coefficient includes feedback from slaved harmonics as well as any original cubic nonlinearity. The resulting amplitude equation, after naming its linear control coefficient $\mu$, isHere $\lambda (k)$ is the critical linear dispersion relation for growth. The condition $g>0$ gives a supercritical bifurcation; $g<0$ calls for higher-order saturation, for example the quintic real Ginzburg-Landau equation.

A concrete check is the cubic Swift–Hohenberg equation $u_t={\epsilon }^2\mu u-(1+{\partial }_x^2{)}^{2}u-g_0{u}^3$. With $u=\epsilon (A{e}^{ix}+\overline{A}{e}^{-ix})+\cdots$, its linear operator contributes $4A_{XX}$ at resonant order, while the cubic resonance is $3|A{|}^{2}A$. Hence $A_T=\mu A+4A_{XX}-3g_0|A{|}^{2}A$. This directly realizes the perturbation-theory coefficients above.

Symmetry independently constrains the form. A fast spatial translation rotates $A$ by a constant phase, so every nondifferential monomial must transform like $A$. The leading allowed nonlinear term is $|A{|}^{2}A$, and a quadratic term is forbidden. A spatial reflection maps $A(X)$ to $\overline{A}(-X)$; it makes the leading coefficients real and rules out a drift term in a reflection-symmetric rest frame. The vanishing first derivative of the linear growth rate at $k_c$ is consistent with the absence of a leading first spatial derivative. These arguments constrain terms but do not calculate $\xi ,g$ or replace the solvability calculation. Breaking reflection or studying an oscillatory instability generally leads to the complex Ginzburg–Landau equation with genuinely complex coefficients.

The principal steady solutions are the zero state and the Ginzburg-Landau plane wavesTheir physical carrier wavenumber is $k_c+\epsilon q$. For $q=0$ the constant phase is arbitrary, giving a circle of uniform-amplitude states. The zero state has Fourier growth rate $\mu -\xi p^2$ and is stable for $\mu <0$. For $g>0$, each existing plane wave is stable to spatially homogeneous amplitude perturbations, with radial eigenvalue $-2g{R}^2$, and neutral to a constant phase shift.

Steady modulated states are also possible. Writing $A=a(X)e^{i\varphi (X)}$ gives the conserved spatial phase current $j=a^2{\varphi }_X$ andIts first integral is $\xi a_X^2/2+\mu a^2/2-g{a}^4/4+\xi j^2/(2a^2)=E$. Boundary conditions select nonuniform periodic amplitudes, fronts or defects from this spatial dynamics. In the invariant real subspace, for example,is a steady kink joining opposite real states. It is not stable in the unrestricted complex equation: the imaginary perturbation $v=\mathrm{sech}(\sqrt{\mu /(2\xi )}X)$ has eigenvalue $\mu /2>0$ under $v_T=\xi v_{XX}+(\mu -g{A}^2)v$. This illustrates why a real-amplitude calculation can miss a phase instability. With real coefficients the Lyapunov functionalsatisfies $A_T=-\delta \mathcal{F}/\delta \overline{A}$ and $d\mathcal{F}/dT=-2\int |A_T{|}^{2}dX$ under compatible boundary conditions. It supplies a variational structure absent from a generic complex-coefficient model.

To test full linear stability of a plane wave, set $A=e^{iqX}(R+u+iv)$ with real perturbations $u,v$. Let $b=g{R}^2=\mu -\xi q^2>0$. The linear equations areFor perturbation wavenumber $p$, the sideband spectrum of a real Ginzburg-Landau plane wave isThe phase branch at small $p$ isTherefore the robust stable band on the infinite line is $q^2<\mu /(3\xi )$, narrower than the existence band $q^2<\mu /\xi$. Crossing its boundary gives the longitudinal Eckhaus instability. At the boundary the quadratic phase-diffusion coefficient vanishes and the leading nonzero term is negative of order $p^4$. More exactly, a nonzero sideband grows only when $p^2<6q^2-2\mu /\xi$; a finite periodic domain for the wave envelope may exclude that interval. Constant-amplitude stability is understood modulo its neutral global phase, and the allowed perturbation wavenumbers must always be specified.

In a second spatial dimension, an anisotropic medium that selects one roll orientation permits an equation with ${\xi }_x{A}_{XX}+{\xi }_y{A}_{YY}$. For positive coefficients its plane-wave analysis is similar, with the second-dimensional contribution adding damping. But simply replacing $A_{XX}$ by an isotropic Laplacian is not a faithful extension for rotationally invariant stationary rolls. There the critical wavevectors form a circle. Near $(k_c,0)$,This forces the anisotropic transverse scaling of an isotropic roll envelope: $X=\epsilon x$, $Y={\epsilon }^{1/2}y$, $T={\epsilon }^{2}t$. For the positive carrier convention used here the Newell–Whitehead–Segel equation isThe opposite fast-carrier convention reverses the accompanying imaginary sign and detuning convention. For a transverse phase perturbation of $A=R{e}^{iqX}$ our convention gives $\sigma =-\xi q{p}^2/k_c-\xi p^4/(4k_c^2)$. Thus $q<0$ permits the transverse zigzag instability, in addition to the longitudinal Eckhaus instability. The fourth transverse derivative and mixed derivative in the square encode the critical-circle geometry.

One complex amplitude still describes only a narrow cone of nearby roll orientations. Competing well-separated wavevectors require coupled amplitude equations; resonant triples can admit quadratic interactions and produce the hexagonal convection amplitude equations when the underlying sign symmetry allows them. Strong disorder, defect cores with large gradients, far-from-onset patterns, boundaries, mean flows and conserved fields may all lie outside the single-amplitude approximation. The method of multiple scales gives controlled leading behavior for small amplitudes and slow modulations, rather than a universal model of all two-dimensional pattern selection.

Assume $\alpha >0$, so a finite nonzero critical wavenumber exists. The linear growth rate of a Fourier mode is $\lambda (k)=-\alpha +\mu k^2-k^4$. Maximizing over $k$ and setting that maximum to zero givesSet $K=k_c^2=\sqrt{\alpha }$. The resonant three-wave triad has wavevectors satisfy ${\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3=0$ and ${\mathbf{k}}_i\cdot {\mathbf{k}}_j=-K/2$ for $i\ne j$. With the slow time and amplitude scaling, every growth, quadratic and cubic term first enters at order ${ϵ}^3$. Projection onto each critical mode removes the nonresonant correction ${\Theta }_3$ by the method of multiple scales solvability condition.

The parameter displacement contributes ${\mu }_{2}KA$ to the first mode. For the quadratic term, use $\nabla \cdot ({\Theta }_1\nabla {\Theta }_1)=\frac{1}{2}{\nabla }^2({\Theta }_1^2)$. The resonant coefficient of ${\Theta }_1^2$ at ${\mathbf{k}}_1$ is $2\bar{B}\bar{C}$, so its contribution is $+\gamma K\bar{B}\bar{C}$.

For the cubic gradient term, an ordered triple of modes with wavevectors $\mathbf{p},\mathbf{q},\mathbf{r}$ summing to ${\mathbf{k}}_1$ contributes $(\mathbf{p}\cdot \mathbf{q})({\mathbf{k}}_1\cdot \mathbf{r})$ times its amplitude product. The three self-interaction permutations yield $-3K^2|A{|}^{2}A$. The six permutations of $({\mathbf{k}}_1,{\mathbf{k}}_2,-{\mathbf{k}}_2)$ yield $-2K^2-4({\mathbf{k}}_1\cdot {\mathbf{k}}_2{)}^2=-3K^2$, and likewise for the third mode. Therefore the hexagonal convection amplitude equations areFor a nonzero convection roll, $B=C=0$ and $|A{|}^2={\mu }_2/(3K)$, requiring ${\mu }_2>0$. Choose its phase so $A=a>0$. The radial perturbation decays, but a perturbation in the two other orientations givesThe eigenvalues are $\pm |\gamma |Ka$. ThusThis is roll instability from broken up-down symmetry. A growing homogeneous amplitude perturbation already proves instability, without a sideband calculation. The source's slightly asymmetric setting means nonzero quadratic coupling. If $\gamma =0$ is allowed literally, the displayed leading equations instead have a continuous sphere of constant-total-intensity states and neutral orientation perturbations. They then establish marginality, not strict growing-mode instability; higher-order effects would decide further selection. Likewise $\alpha =0$ would invalidate the assumed finite-$k_c$ triad scaling.

In the hexagonal convection amplitude equations, a pure convection roll has $|A{|}^2={\mu }_2/(3K)$ and $B=C=0$. The transverse diagonal linear growth cancels because self and cross cubic coefficients coincide. Broken up-down symmetry leaves the off-diagonal coupling $\gamma KA$, giving eigenvalues $\pm |\gamma |K|A|$ and an unstable orientation perturbation whenever $\gamma \ne 0$. With zero quadratic coupling these directions are neutral at the retained order, so that degenerate case cannot be declared strictly unstable by this calculation.