Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 54 (Symmetries and Patterns)

Let $q_j$ denote the four critical wavevectors, and write $A_j=R_j{e}^{i{\theta }_j}$. A spatial translation by $(s,t)$ acts on the amplitudes asThe two independent translation directions in phase space are $(2,2,1,1)$ and $(1,-1,2,-2)$. Their common orthogonal complement is spanned by $(2,-2,-1,1)$ and $(1,1,-2,-2)$. Hence translation invariants of Fourier-mode phases can be chosen asAway from zero amplitudes, a translation-invariant function of the phases is a function of these two invariant angles. The four magnitudes are themselves invariant. Thus an equivariant dynamical system on the eight-real-dimensional centre manifold descends, after quotienting translation symmetry, to four magnitude equations and two invariant-phase equations. The phase coordinates are singular where an amplitude vanishes; the original complex-amplitude equations remain smooth there.

The wavevector selection rule for equivariant monomials proves the cubic regularity. For a monomial ${\prod }_j{A}_j^{p_j}{\overline{A}}_j^{q_j}$ in the first equation, put $d_j=p_j-q_j$. Translation covariance requiresAt total degree $N$, additionally $\sum |d_j|\le N$ and $N-\sum |d_j|$ is even. Conversely, any integer vector satisfying these conditions yields a monomial, with any remaining even degree supplied by factors $|A_j{|}^2$. This makes the degree test exhaustive. For example, eliminate the first two entries:Checking the integer pairs $|d_3|+|d_4|\le N$, retaining only integral $d_1,d_2$ with the norm and parity conditions, givesAt degree three the only possibility is therefore $A_1|A_j{|}^2$, $j=1,\dots ,4$. Square symmetry gives the same conclusion for every other equation. All cubic terms are regular.

Two generators of the square dihedral group act as follows. Reflection in the $x$ axis gives $(A_1,A_2,A_3,A_4)\mapsto (A_2,A_1,A_4,A_3)$, while interchange of $x,y$ gives $(A_3,{\overline{A}}_4,A_1,{\overline{A}}_2)$. Rotation by $\pi$ conjugates every amplitude, forcing the coefficients in a steady normal form of a dynamical system to be real. With real unfolding parameter $\sigma$ and real coefficients $c_1,c_2,c_3,c_4$, defineThe most general cubic normal form is ${\dot{A}}_j=A_j{G}_j+O(|A{|}^5)$. There are no further coefficient identifications: the square-group stabilizer of one of these oblique wavevectors is trivial, so its three couplings to the other magnitudes are independent. At cubic order ${\dot{R}}_j=R_j{G}_j$ and all phases, including ${\chi }_1,{\chi }_2$, are constant.

The degree-five row of the wavevector selection rule for equivariant monomials in part (a) has two vectors other than $(1,0,0,0)$. They correspond exactly toThe vector $(1,0,0,0)$ gives only regular fifth-order terms $A_1|A_j{|}^2|A_k{|}^2$. Omitting those as required, and applying the same two square-symmetry generators, the truncated normal form of a dynamical system isHere $e,f$ are real because rotation by $\pi$ acts by complex conjugation. These are equations for steady patterns, so all four complex right-hand sides must vanish; constancy of the two invariant angles alone would only establish a relative equilibrium and could allow translation drift.

Take a nonzero equal-magnitude state, $|A_j|=R>0$, and set $z=R^2$, $S=c_1+c_2+c_3+c_4$, $\rho =\sigma +Sz$. Dividing each equation by its nonzero amplitude gives, in order,Taking imaginary parts, adding and subtracting pairs, yieldsThus, unless $e=f=0$, each invariant phase is zero or $\pi$ modulo $2\pi$. Equality of the real parts further givesThis identifies the equal-amplitude states of an eight-mode square pattern completely. For generic $e\ne f$, the two nonzero phase types, with their amplitude equations, areRepresentatives are respectively $(A_1,A_2,A_3,A_4)=R(1,1,1,1)$ and $R(-1,1,-1,1)$. Spatial translations change individual phases but leave the indicated phase pair fixed. Reflection acts on the invariant angles as $({\chi }_1,{\chi }_2)\mapsto (-{\chi }_1,{\chi }_2)$, and interchange of $x,y$ as $({\chi }_1,{\chi }_2)\mapsto (-{\chi }_2,-{\chi }_1)$. Thus the two rows are not equivalent under square symmetry and translations. Every positive root of the corresponding polynomial gives an amplitude branch of that type; if there is no positive root, that type is absent at those parameter values.

There are two coefficient degeneracies to include. When $e=f\ne 0$, the mixed pairs $(0,\pi )$ and $(\pi ,0)$ are allowed and are equivalent under square symmetry. They form one additional phase type, with amplitude equation $\boxed{\sigma +Sz=0}$; $iR(1,1,1,1)$ is a representative. When $e=f=0$, all invariant phases are unrestricted, and the same cubic amplitude equation gives a continuous family modulo translations and the finite square-group action. Even if $e+f=0$ but $e\ne f$, only the original two phase types remain: the equality of their amplitude equations does not remove the phase constraints. Finally, the all-zero state is always a separate steady state and has no defined polar phases.

The spatial mean is conserved: averaging every term in the partial differential equation over the periodic cell gives ${\partial }_t⟨T⟩=0$. Adding a constant to $T$ also preserves the equation. We therefore fix the mean, conveniently at zero. This qualification is essential: without it the constant Fourier mode contributes an additional neutral direction, so the centre subspace would have five real dimensions rather than four.

On a Fourier mode $e^{i(mx+ny)}$, the linearized linear operator has eigenvalueOn the fixed-mean subspace, the first eigenvalues to reach zero are those with $m^2+n^2=1$. At $\mu =1$ these are the modes $e^{\pm ix},e^{\pm iy}$; every other nonconstant Fourier mode is strictly damped. The reality condition leaves two complex amplitudes, hence a four-real-dimensional centre manifold on the fixed-mean subspace. The trivial state loses linear stability as $\mu$ increases through one.

Use slow time $\tau ={\epsilon }^{2}t$, and write $T=\epsilon W+{\epsilon }^3{T}_3+\cdots$, whereThere is no quadratic nonlinearity, so no forced second-order correction is needed. At order ${\epsilon }^3$, with $\mu =1+{\epsilon }^2\nu$, projection onto the critical Fourier modes givesThe correction $T_3$ does not contribute to the critical projection because the linear operator vanishes on those modes. To find the nonlinear coefficient of $e^{ix}$, split the divergence term intoThe first term contributes $-3|A{|}^{2}A$. In the second, only the zero-$y$ Fourier coefficient of $W_y^2$ contributes; it is $2|B{|}^2$, and differentiating $W_x$ once more gives $-2|B{|}^{2}A$. The third term has zero $y$-average, and the fourth has no $x$-dependent critical component. Interchanging $x,y$ gives the other amplitude equation. Thus the conserved-mean convection amplitude equations areTo establish the selected planform, put $A=r{e}^{i\theta }$, $B=s{e}^{i\psi }$. The phases are constant and the magnitudes obeyFor $\nu >0$, the square state has $r=s=\sqrt{\nu /5}$. Its magnitude Jacobian matrix is $-(2\nu /5)\left(\begin{array}{cc}3& 2\\ 2& 3\end{array}\right)$, with eigenvalues $-2\nu$ and $-2\nu /5$. Its two neutral phases correspond to translation symmetry. The pure roll has $r^2=\nu /3,s=0$; an infinitesimal $B$ disturbance grows at rate $\nu -2\nu /3=\nu /3>0$. The other pure roll is equivalent, while the trivial state is unstable for $\nu >0$. Consequently squares, with $|A{|}^2=|B{|}^2=\nu /5$, are the stable small-amplitude planform, modulo translations. For $\nu <0$ the trivial state is stable.

Let $F(T)$ denote the right-hand side of the temperature partial differential equation. For a periodic variation $\eta$, integration by parts givesAlong the evolution, $T_t=F(T)$, so this is a gradient flow with increasing functional:For $s=|\nabla T{|}^2$, completing the square givesThus the increasing-gradient functional for poorly conducting convection converges to a finite limit along any globally smooth solution, andIn particular, there are arbitrarily late times at which $\Vert T_t{\Vert }_{L^2}$ is arbitrarily small. A nonstationary time-periodic solution is impossible: it would have a strictly positive increase of $V$ over a temporal period. More generally, if the orbit is precompact in a topology strong enough for the equation, its limiting states are stationary. With the conserved mean fixed, the displayed square completion bounds $\Vert \Delta T{\Vert }_{L^2}$, and periodic elliptic estimates give an $H^2$ bound; the usual smoothing and compactness for a globally regular parabolic solution support this limiting-state conclusion. The inequalities alone do not prove convergence to one uniquely specified equilibrium, and stationary families related by translation symmetry can remain.

Now take a smooth, nonconstant periodic roll $T_0(x)$. Multiplying its stationary partial differential equation by $T_0$ and using periodic integration by parts yieldsThe primes here are essential: the first and fourth terms involve $T_0^{\prime }$, not $T_0$. Write $a=T_0^{\prime }$, $b=T_0^{\prime \prime }$, $M_2=⟨a^2⟩$, $M_4=⟨a^4⟩$, and $N_2=⟨b^2⟩$. The orthogonal-roll perturbation hasThe cross term in the squared Laplacian averages to zero since $⟨b⟩=0$. Independence of the $x,y$ cell integrals then gives the exact polynomial differenceIts leading coefficient is half the variance of $a^2$ over the period. That variance is strictly positive: a nonconstant smooth periodic function has a derivative which vanishes somewhere and is nonzero somewhere, so $a^2$ is not constant. Hence arbitrarily small transverse perturbations increase $V$ above the roll value.

One can turn this energy comparison into linear instability. The linearized evolution at the roll is the self-adjoint operatorChoose the fixed-mean perturbation $\eta =T_0(y)-⟨T_0⟩$. Since $D^{2}V[T_0](\eta ,\eta )=⟨\eta L\eta ⟩=M_4-M_2^2>0$, the Rayleigh quotient is positive. The periodic self-adjoint fourth-order operator therefore has a positive eigenvalue. This proves the transverse energy instability of nonconstant temperature rolls: every nonconstant roll is unstable wherever it exists.

The existence qualification matters. Constants are also stationary and are stable on the fixed-mean subspace for $0<\mu <1$. Moreover, the periodic Poincaré inequality gives $N_2\ge M_2$, so the roll identity implies $\mu M_2=N_2+M_4>M_2$ for a nonconstant $2\pi$-periodic roll. Such rolls require $\mu >1$ on this specified cell; the conclusion for all positive $\mu$ is conditional on a nonconstant roll being available.

The spatial reflection $x\mapsto -x$ acts on the two complex amplitudes as $(A,B)\mapsto (A,-B)$, because the first spatial eigenfunction is even and the second is odd. The first equation must therefore be even in $B,\overline{B}$, and the second odd. For a nonresonant double Hopf bifurcation, the temporal phases give independent actions $A\mapsto e^{i{\phi }_1}A$, $B\mapsto e^{i{\phi }_2}B$ in the cubic normal form of a dynamical system. A cubic monomial in the first equation must have net $A$ phase weight one and net $B$ phase weight zero. Only $A|A{|}^2,A|B{|}^2$ satisfy this; the second equation similarly has $B|B{|}^2,B|A{|}^2$. Parametrizing the independent cross-saturation coefficients givesAll coefficients can be complex: their real parts determine amplitude growth and their imaginary parts frequency shifts. These are cubic truncations of a family of reduced vector fields on the four-real-dimensional extended centre manifold, with the two unfolding parameters entering ${\nu }_1,{\nu }_2$.

The nonresonance assumption must be made explicit. Merely having ${\omega }_1\ne {\omega }_2$ does not imply this form: for ${\omega }_1=2{\omega }_2$, the quadratic terms $B^2$ in $\dot{A}$ and $A\overline{B}$ in $\dot{B}$ satisfy both temporal resonance and spatial parity. Thus the initial form is the generic nonresonant double-Hopf form, rather than a statement for every possible unequal frequency pair.

At the one-to-one reflection-symmetric one-to-one Hopf resonance, only the common temporal phase action remains. Cubic terms must then have total phase weight one. Spatial reflection retains, in the first equation, the three terms $A|A{|}^2,A|B{|}^2,B^2\overline{A}$; the second retains $B|B{|}^2,B|A{|}^2,A^2\overline{B}$. Common phase covariance excludes all quadratic terms. The additional resonant terms are consequently $-{\gamma }_1{B}^2\overline{A}$ and $-{\gamma }_2{A}^2\overline{B}$, with independent complex coefficients. This is the reflection-symmetric one-to-one Hopf resonance.

In the specified equal-coefficient specialization, put $r=R^2$. A nonzero pure-$A$ periodic branch exists for $\mathrm{Re}{\nu }_1>0$ and hasIts radial eigenvalue is $-2{\alpha }_{R}r<0$, and its phase is neutral. For a transverse $B$ perturbation the linear equation isIts time dependence can be removed exactly by $B=C{e}^{i\Omega t}$. Using ${\nu }_1-\alpha r=i\Omega$ and ${\nu }_2={\nu }_1-d$ givesWriting $C=u+iv$, $d=d_R+i{d}_I$, $\beta ={\beta }_R+i{\beta }_I$ yields the constant real stability matrixLet $Q=\mathrm{Re}(\beta \overline{d})={\beta }_R{d}_R+{\beta }_I{d}_I$. The transverse characteristic polynomial isBoth transverse eigenvalues have negative real part exactly whenThese hold at $r=0$ because $d_R>0$. Thus a sufficiently small pure branch is transversely stable; it is stable modulo its neutral temporal phase until one of the two inequalities fails. The secondary steady and Hopf thresholds of a resonant pure mode are derived in the two parts below. They refer to accessible points on the pure branch; in a multi-parameter problem $d$ may also vary, so a proposed parameter path must actually cross the corresponding surface.

A steady transverse bifurcation requires a zero eigenvalue, hence $\det M=0$. This givesThe inequality is required for a positive amplitude. Since $d_R>0$, the numerator cannot vanish. The zero eigenvalue is simple provided the other eigenvalue, the trace $-2(d_R+{\beta }_R{r}_S)$, is nonzero. Equivalently,With $d,\beta$ fixed, ${\partial }_r\det M=2Q\ne 0$, so the simple zero eigenvalue crosses transversely. If this is to be the first instability of the stable pure branch, its other eigenvalue must still be negative. Comparing with $r_H=-d_R/{\beta }_R$ gives the stronger conditionFor $Q_{\ast }<Q<0$, a simple steady threshold still exists, but it lies after the Hopf threshold and the pure branch is already unstable there. At $Q=Q_{\ast }$ both transverse eigenvalues vanish, so the event is a double-zero degeneracy rather than an ordinary simple steady bifurcation.

The reduced system has the reflection symmetry $C\mapsto -C$. At a generic simple steady threshold, the new mixed states therefore occur in a reflection-related pair, with small nonzero $B$ and a corresponding correction to the $A$ magnitude and common frequency. In the original variables they are phase-locked periodic solutions involving both spatial modes: $A=a{e}^{i\tilde{\Omega }t}$, $B=b{e}^{i\tilde{\Omega }t}$, with constant nonzero complex $a,b$. The odd spatial component breaks the reflection symmetry of the pure even mode; the two branches are exchanged by $b\mapsto -b$. Determining whether the branches are supercritical or subcritical, and their stability, requires the reduced nonlinear coefficient and is not determined by this linear threshold alone.

A Hopf crossing of the transverse stability matrix requires zero trace and strictly positive determinant. Since ${\beta }_R<0$, the trace condition gives the positive thresholdThe determinant there isThus the required coefficient condition and the rotating-frame Hopf frequency areThe trace derivative $-2{\beta }_R$ is strictly positive, so with $d,\beta$ fixed the real parts cross zero with nonzero speed. Under this condition the determinant stays positive from $r=0$ to $r_H$; the pure branch first loses linear stability through this Hopf bifurcation. When $Q<Q_{\ast }$ its first instability is instead the steady one of part (a), and $D_H<0$ prevents a Hopf pair at the trace-zero point. Equality gives the already noted double-zero degeneracy.

Generically the new branch is periodic in the symmetry-reduced rotating frame, with a small oscillating odd-mode amplitude and a modulation of the even-mode amplitude. Reconstructing the common temporal phase combines the modulation frequency with the carrier frequency. The original amplitudes therefore describe modulated periodic oscillations, generically a two-frequency quasiperiodic state. If the two frequencies are rationally related, the reconstructed motion can be periodic instead. The spatial pattern no longer has a fixed mixture of the even and odd modes: their relative amplitudes and phase oscillate. As with the steady case, nonlinear nondegeneracy is needed for an ordinary Hopf branch, and its criticality or stability cannot be inferred from the trace and determinant alone.

A periodic pattern breaks continuous translation symmetry: translating it changes its phase without changing its shape or its energy. Consequently a uniform phase change is neutral. A slowly varying phase change alters the local spacing or orientation and can grow even while ordinary uniform amplitude perturbations decay. These are long-wavelength sideband instabilities. The relevant wavelengths are long compared with the carrier pattern, so their leading dynamics is often a phase modulation equation. Positive phase diffusion smooths spacing errors; negative phase diffusion amplifies them. Near onset, an amplitude equation computes this diffusion explicitly. Away from onset, one must calculate it from the full nonlinear periodic pattern.

For a supercritical stationary pattern in one dimension, rescale the real Ginzburg–Landau equation toA detuned periodic pattern is represented by $A=R{e}^{iqx}$, with $R^2=\mu -q^2>0$. Perturb its magnitude and phase by $A=(R+u)e^{i(qx+\varphi )}$. Linearization givesFor a Fourier mode proportional to $e^{\lambda t+ikx}$, the two eigenvalues are the sideband spectrum of a real Ginzburg-Landau plane wave:The magnitude branch remains damped at small $k$. The phase branch expands asHence the Eckhaus-stable band is $\boxed{q^2<\mu /3}$, narrower than the existence band $q^2<\mu$. This Eckhaus instability is longitudinal: it compresses and dilates the pattern along its wavevector. The coupling to the relaxing magnitude is crucial; examining phase diffusion while holding the magnitude fixed would miss the destabilizing $2q^2/R^2$ correction. At the stability boundary the $k^2$ coefficient vanishes, and the next term is negative, $-2q^4{k}^4/R^6$. Beyond the boundary, a phase distortion can create amplitude zeros and phase slips, changing the number of periods. These nonlinear events are possible outcomes, not conclusions of the linear calculation.

A fixed periodic domain allows only discrete sideband wavenumbers. Indeed, the displayed spectrum is unstable only if $0<k^2<6q^2-2\mu$. A domain too short to support such a Fourier mode can suppress the nominal long-wave instability. Thus an infinite-domain stability band and a finite-domain threshold must be distinguished.

The zigzag instability concerns transverse bending of rolls in an isotropic two-dimensional system. Let $k_c$ be the preferred carrier wavenumber, use an $x$-directed carrier, and denote its slow longitudinal detuning by $q$. Rotational covariance near onset leads to the anisotropic envelope equationThe derivative combination comes from expanding $|\mathbf{k}|-k_c$: a small transverse carrier component changes the length of the wavevector only at second order. A straight-roll envelope $A=R{e}^{iqX}$ has $g{R}^2=\mu -\xi q^2$. For a purely transverse phase perturbation $A\simeq R{e}^{iqX}(1+i\varphi (Y,t))$, magnitude and phase decouple at linear order, andThe transverse growth rate is thereforeRolls with $q<0$, whose total wavenumber is below the preferred value, are zigzag-unstable to sufficiently long transverse modulations. Bending increases their local wavevector length and can relieve this mismatch. Rolls with $q>0$ are transversely stable in this near-onset model; at $q=0$ the restoring force first appears at fourth spatial order. Combining this with the longitudinal Eckhaus instability gives the robust interior stable band $0<q<\sqrt{\mu /(3\xi )}$ for this simplest isotropic roll model. Anisotropy, a pinned orientation, additional mean fields, or other pattern symmetries can modify this conclusion.

For oscillatory patterns the coefficients of the amplitude equation are generally complex. Consider the scaled complex Ginzburg–Landau equationwhere $c_1$ measures linear dispersion and $c_3$ the amplitude-dependent frequency shift. The uniform oscillation is $A=e^{-i{c}_{3}t}$. Put $A=e^{-i{c}_{3}t}(1+u+iv)$, with real infinitesimal $u,v$. The linearized magnitude and phase equations areTheir characteristic polynomial for a sideband of wavenumber $k$ isThe magnitude mode has leading eigenvalue $-2$, while the neutral phase mode hasThus the Benjamin-Feir instability occurs for $\boxed{1+c_1{c}_3<0}$; the complementary Benjamin-Feir stability condition is $1+c_1{c}_3>0$. Both dispersive couplings matter: the phase perturbs the magnitude through $c_1$, and the magnitude feeds back into phase through $c_3$. In the unstable regime, higher spatial derivatives regularize sufficiently short disturbances. On a finite domain, an unstable mode must satisfy $0<k^2<-2(1+c_1{c}_3)/(1+c_1^2)$. At equality of the phase-diffusion coefficient to zero, the leading phase damping is fourth order, not exponential growth at order $k^2$.

Oscillatory plane waves with a nonzero carrier detuning also have a generalized Eckhaus boundary. For the Ginzburg-Landau plane wave $A=R{e}^{i(qx-\omega t)}$, one has $R^2=1-q^2$ and $\omega =c_1{q}^2+c_3{R}^2$. Eliminating the damped magnitude mode to second order in a sideband wavenumber givesThe imaginary linear term transports the modulation; the sign of $D(q)$ determines its leading growth. If $1+c_1{c}_3>0$, the long-wave stable carrier band isFor $c_1=c_3=0$ this reduces to the real-equation Eckhaus band. If $1+c_1{c}_3<0$, even the zero-detuning oscillation has the Benjamin-Feir instability. Nonlinear outcomes can include smooth phase modulation, defect production or phase turbulence, depending on the equation and parameters.

The general nonlinear-pattern problem starts from a periodic solution $u_0(x)$ of the full field equations, not from a small-amplitude envelope. Linearize about it to obtain an operator $L$ with periodic coefficients. Write a disturbance as $e^{\lambda t+iq\cdot x}v(x)$ with $v$ periodic on the pattern cell. The Bloch stability operatoracts on this fixed cell and determines the entire sideband spectrum. Translation covariance gives $L(0){\partial }_x{u}_0=0$ in one dimension, and corresponding neutral directions for each independent broken translation in several dimensions. Oscillatory base patterns require the analogous Floquet theory calculation over their temporal period, or a rotating frame where applicable.

For a simple translational zero eigenvalue, the phase-diffusion calculation can be made explicit. Expandwhere $v_0=u_0^{\prime }$ and $L_0{v}_0=0$. Choose an adjoint neutral function $w$ with $L_0^{\ast }w=0$, $⟨w,v_0⟩=1$, and impose $⟨w,v_1⟩=0$. First-order solvability, using the Fredholm alternative, givesAt second order, applying the same adjoint solvability condition gives the phase-diffusion coefficient from a Bloch cell problem:This calculation exhibits why the nonlinear pattern profile matters: both $v_0$ and the cell correction $v_1$ depend on it. A negative $D$ signals a long-wave instability provided the rest of the spectrum is stable. With several phase directions, or an extra neutral conserved mean, one obtains a coupled modulation system and diffusion matrices rather than this scalar formula. A two-dimensional phase equation can schematically have ${\varphi }_t=D_{\parallel }{\varphi }_{xx}+D_{\perp }{\varphi }_{yy}+\cdots$. Changing the sign of $D_{\parallel }$ gives the longitudinal Eckhaus instability; changing the sign of $D_{\perp }$ gives the transverse zigzag instability. For oscillatory waves, nonvariational magnitude-phase coupling gives the Benjamin-Feir instability.

These mechanisms extend beyond onset, but the near-onset numerical boundaries do not: their coefficients must be replaced by those obtained from the actual nonlinear pattern and its cell problems. Long-wave linear stability is also only one part of stability; finite-wavenumber eigenvalues, additional modes, defects, and boundary constraints can intervene. The unifying calculation is the splitting of translation-induced neutral modes at small modulation wavenumber, together with the coupling to the relaxing magnitude and any other slow fields.