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The velocity coupling changes the momenta and the undifferentiated field terms. Variation with respect to ${\varphi }_2$ gives the equation for ${\varphi }_1$:so the two Euler-Lagrange field equations areThey describe damped and antidamped fields from an antisymmetric velocity coupling. With the convention ${\varphi }_j=A_j{e}^{i(k_{j}x-{\omega }_{j}t)}$, their characteristic equations areThus the dispersion relations and modes areFor each spatial Fourier mode, the general solution is the linear combination of its two temporal branches; spatial superposition gives the general field. Take real parts or suitable complex conjugate pairs to obtain the required real fields. At the repeated-root threshold, the independent second temporal solution contains an extra factor $t$, as detailed next.

Freeze a uniform state with $0<\alpha <1$, ${\rho }_i>0$, and seek a finite characteristic speed $c$. Denote its perturbation amplitudes by $a,v_1,v_2,\pi$. The two phase-mass equations and two momentum equations giveEliminating the amplitudes givesWith $s^2={\rho }_1(1-\alpha )/({\rho }_2\alpha )>0$, factor the quadratic over the complex numbers to obtainFor nonzero relative velocity the speeds are a nonreal conjugate pair. A Fourier perturbation $e^{ik(x-ct)}$ on the growing branch has magnitude $e^{|k||\mathrm{Im}c|t}$. Arbitrarily short waves therefore grow arbitrarily quickly, preventing a continuous dependence estimate in the usual finite-order Sobolev norms. This generic slip state is ill-posed as an initial value problem.

The pressure has no independent time-evolution equation; it enforces the incompressible volume constraint, so the request concerns the finite characteristic speeds rather than a freely propagating pressure mode. At $u_1=u_2$ the two speeds coincide and this particular nonreal-speed argument disappears; further degeneracy analysis is needed there, rather than calling coincident real speeds complex. The issue is complex characteristic speeds of a pressure-coupled two-fluid model.

Multiply the complex Fourier series by $e^{-i\omega rt}$ and integrate over one period. Fourier orthogonality removes every term except the one with index $r$, givingFor general square-integrable functions, this is an orthogonal projection onto the corresponding Fourier mode. It remains valid when the expansion converges only in L2 space: integrate finite Fourier partial sums first, then pass to the limit using the Cauchy-Schwarz inequality. No unjustified pointwise exchange of an infinite series and an integral is needed.

Let ${\hat{u}}_0$ be the Fourier transform of the initial temperature. Solving the scalar linear ordinary differential equation givesRecover $u$ using the Fourier inversion theorem, with inverse factor $(2\pi {)}^{-2}$ for this two-dimensional convention. Equivalently, take the convolution of the initial data with the two-dimensional heat kernel. On a periodic square, the same exponential applies to each discrete Fourier mode, with $k_x=2\pi m/L_x$ and $k_y=2\pi n/L_y$.

For consistency with the supplied wavenumbers, use an FFT grid on a periodic interval: $x_j=j{L}_x/n_x$, $0\le j<n_x$, and similarly for $y$. In MATLAB, replace the endpoint-inclusive coordinate definitions by x = (0:nx-1)*Lx/nx; and y = (0:ny-1)*Ly/ny;. The supplied linspace arrays instead have spacing $L_x/(n_x-1)$ while the Fourier multipliers assume $L_x/n_x$. Including both copies of the periodic endpoint creates a grid inconsistency, separate from high-frequency aliasing.

Diffusion smooths and broadens the hot spot while conserving its integral. For an isolated two-dimensional initial profile $u_0=e^{-r^2/a}$, the Gaussian hot spot under diffusion isHere $a=0.1\mathrm{c}{\mathrm{m}}^2$ and $t=10\mathrm{s}$, giving a width parameter of $0.5\mathrm{c}{\mathrm{m}}^2$ and a peak of $0.20$ in the initial temperature units. The periodic-grid calculation agrees to displayed precision because the periodic copies are far away. Running the endpoint-inclusive listing as written instead gives a sampled initial peak near $0.9695$ and final peak near $0.1963$; the difference is consistent with the coordinate-grid issue, rather than a different diffusion law.

Increasing positive thermal diffusivity at fixed time produces a wider, lower peak. Each nonconstant Fourier mode decays faster, while the spatial mean of a periodic heat equation remains fixed. The profiles approach a uniform mean at long times under these periodic boundary conditions.

Aliasing occurs because a sampled grid cannot distinguish certain different continuous Fourier modes. With spacing $h$, modes $k$ and $k+2\pi m/h$ agree at every sample point. Frequencies above the Nyquist frequency can therefore be misidentified as lower frequencies, changing the resolved spectrum and its subsequent diffusion rate.

For the corrected grid, $h=10/128\mathrm{cm}$ and the maximum resolved angular wavenumber is approximately $\pi /h\simeq 40.2\mathrm{c}{\mathrm{m}}^{-1}$. The initial Gaussian function has spectral envelope proportional to $e^{-a|k{|}^2/4}$. At that one-coordinate cutoff the factor is about $e^{-40.4}$, so the omitted high-frequency tail is tiny. Its continuation mismatch at the distant edges is also tiny. Significant aliasing is not expected for this resolved, smooth initial condition.

The linear constant-coefficient heat equation only damps existing modes; it does not generate higher frequencies through nonlinear products. An unresolved narrow peak, a sharp edge or a nonlinear source would require a separate resolution assessment. The duplicated-endpoint grid issue described above should not be diagnosed as high-frequency folding.

A periodic domain identifies opposite faces of a fundamental cell, producing a flat torus. On an interval $[0,L]$, smooth periodic boundary conditions require $u(0,t)=u(L,t)$ and matching spatial derivatives. These endpoints are one location, rather than two independently controlled walls. Periodic integration by parts has no boundary term, and zero-mean functions have no zero Fourier mode.

Use dimensionless $x$ and $t$, the periodic domain $[0,2\pi ]$, and the same initial conditionWith zero PDE source term, compare the heat equation $u_t=u_{xx}$ and the transport equation $u_t+u_x=0$. Their exact solutions areEach cosine satisfies ${\partial }_x^2\cos (nx)=-n^2\cos (nx)$, so the heat equation reduces its amplitude according to $A_n^{\prime }=-n^2{A}_n$. The superposition principle combines the constant and two Fourier modes. The transport equation shifts the argument of the entire profile. Both formulas meet the initial condition and matching periodic boundary conditions exactly.^[ref]

At $t=1$, the heat amplitudes are approximately $0.368$ and $0.0000494$: the small ripples disappear much sooner than the broad variation. The transport equation retains amplitudes $1$ and $0.4$, shifting the profile right by one unit with wraparound. Both preserve spatial average $1$, but only the heat equation approaches the uniform profile. This gives diffusion its usefulness for smoothing uneven temperatures, while the transport equation models a profile carried by a flow. The signed profile can represent a scaled temperature deviation.

To restore units, let physical position be $X=\ell x$ and time be $T=\tau t$. For $U_T=D{U}_{XX}$, the dimensionless coefficient is $D\tau /{\ell }^2$, with $D$ measured in length squared per time. For $U_T+c{U}_X=0$, it is $c\tau /\ell$, with $c$ measured in length per time. Both plotted coefficients equal one when $D={\ell }^2/\tau$ and $c=\ell /\tau$. Thus the example compares specified models on the same scales, rather than identifying diffusivity with velocity.

The magnitude of the phase velocity of a real-frequency plane Fourier mode is $|\omega |/|\mathbf{k}|$. In an anisotropic medium it depends on the direction of the wave vector and need not equal the group velocity magnitude.

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.

For the Poisson equation $\Delta u={\sum }_{n\ge 1}{b}_n\sin (n\theta )$ on the unit disk with zero Dirichlet boundary condition, separation of variables in polar coordinates gives $r^2{R}_n^{\prime \prime }+r{R}_n^{\prime }-n^2{R}_n=r^2$. For $n\ne 2$, boundedness at $r=0$ and $R_n(1)=0$ select $R_n=(r^n-r^2)/(n^2-4)$. The resonant mode is $R_2=r^2\log r/4$. For discontinuous angular data, the Poisson equation is interpreted in its weak formulation; its Dirichlet energy gives uniqueness. Orthogonality of the angular Fourier modes gives $\int ufdA=-{\sum }_n\pi b_n^2/[4(n+2{)}^2]$, with the resonant case obtained directly or by the limit $n\to 2$.

For an infinite circular cylinder of radius $R$, assume the electrostatic potential is independent of the axial coordinate. An angular Fourier mode $s_n\cos n\theta$, or $s_n\sin n\theta$, in the prescribed jump ${\partial }_r\Phi {|}_{R^{+}}-{\partial }_r\Phi {|}_{R^{-}}=-\sigma (\theta )$ produces the same angular mode in the potential, multiplied byHere $r_{<}=\min (r,R)$ and $r_{>}=\max (r,R)$ in the abbreviated header formula. Separation of variables selects $r^n$ inside by regularity and $r^{-n}$ outside by decay. Continuity equates their boundary values, and the derivative jump is minus $2n/R$ times that value. A nonzero angular mean of the surface charge density introduces a logarithmic term, so zero mean is necessary for a potential decaying at infinity in this two-dimensional reduction.

Fix zero mean and period $P$ in the Kuramoto-Sivashinsky equation. At $P=2\pi$ the first Fourier mode becomes neutral and all higher modes are damped. For $k=2\pi /P$, expand $u=a\sin (kx)+b\sin (2kx)+\cdots$. The first equations are $\dot{a}={\lambda }_{1}a+kab/2+\cdots$, $\dot{b}={\lambda }_{2}b-k{a}^2/2+\cdots$, with ${\lambda }_j=(jk{)}^2-(jk{)}^4$. Slaving the second harmonic gives $\dot{a}={\lambda }_{1}a+k^2{a}^3/(4{\lambda }_2)+\cdots$. Since ${\lambda }_2=-12$ at onset, this is a supercritical bifurcation. The small nonzero branch is stable in the sense of orbital stability against disturbances of the same fixed period, modulo translation; stability against disturbances of larger period does not follow from this local argument.

In a Parker dynamo wave model, an alpha effect of the form $\alpha ={ϵ}^{-1}{\alpha }_0\sin (\omega t/ϵ)$ has zero time average but can generate a finite averaged coupling. The method of multiple scales gives $\tilde{A}=-({\alpha }_0/\omega )\cos (\omega \tau )\overline{B}$ and $\tilde{B}=-(\Omega {\alpha }_0/{\omega }^2)\sin (\omega \tau ){\partial }_x\overline{B}$, where $\tau =t/ϵ$. Consequently $C=\Omega {\alpha }_0^2/(2{\omega }^2)$. With diffusion operator ${\partial }_x^2-{\ell }^2$, the averaged Fourier mode growth exponents are $-k^2-{\ell }^2\pm |k|\sqrt{\Omega C}$. For freely adjustable real $k$, growth requires $\Omega C>4{\ell }^2$. The result is an asymptotic averaged model for fixed nonzero $\omega$, not an exact replacement at finite $ϵ$.

A layer of half-thickness $h_0$, dynamic viscosity $\mu$ and density $\rho$, between semi-infinite fluids of viscosity $\lambda \mu$ and densities $\rho \mp \Delta \rho$, relaxes a symmetric thickness Fourier mode. In the plug-like extensional viscosity regime, mass conservation and $4\mu (h{u}_x{)}_x=h\Delta \rho g{h}_x-{\sigma }_{xz}^{+}$ together with the Fourier traction map for a viscous half-space give the displayed rate, where $\kappa =|k|h_0$. Inner longitudinal extension dominates for $\lambda \ll \kappa$; outer tangential resistance dominates for $\kappa \ll \lambda$ while the plug approximation remains valid.

Three Fourier modes form a spatial resonance when their wavevectors sum to zero. Products of the conjugate second and third modes then force the first mode, with cyclic counterparts. Such terms couple both amplitudes and phases, unlike self-saturation terms proportional only to squared magnitudes. For three equal wavenumbers, the vectors are separated by $120$ degrees and support hexagonal pattern interactions.

In a weakly nonlinear expansion of the long-wave convection equation with broken Boussinesq symmetry at $\mu =2$, a critical Fourier mode ${\Theta }_1=A{e}^{ix}+c.c.$ generates ${\Theta }_2=(2is/9)A^2{e}^{2ix}+c.c.$. The quadratic interaction of these first and second harmonics feeds back $8s^2|A{|}^{2}A/9$ into the critical Fourier mode, while the cubic gradient nonlinearity contributes $-3|A{|}^{2}A$. The Fredholm solvability condition is consequently ${\mu }_{2}A-g|A{|}^{2}A=0$. The sign of $g$ distinguishes supercritical and subcritical branches; $g=0$ requires a higher-order amplitude equation.

For angular Fourier modes of integer order $n\ge 2$, the biharmonic equation in polar coordinates has radial basis $r^n,r^{-n},r^{n+2},r^{2-n}$. Apply the Laplacian twice to obtain the displayed indicial equation. For $n=0$, repeated roots give the radial basis $1,\log r,r^2,r^2\log r$; for $n=1$, the basis is $r^{-1},r,r\log r,r^3$. These are Cauchy-Euler equations. Regularity at the origin and boundary conditions select the admissible powers. For $n=2$, a constant radial factor still multiplies $\cos 2\theta$ or $\sin 2\theta$, so it is bounded but fails continuity at the origin.

For a planar magnetic field written as $\mathbf{B}=\nabla \times (A\hat{\mathbf{z}})$ and a circular velocity $\mathbf{u}=s\Omega (s)\hat{\mathbf{\varphi }}$, the resistive induction equation reduces, after choosing the additive gauge transformation, to ${\partial }_{t}A+\Omega {\partial }_{\varphi }A=\eta {\nabla }^{2}A$. A Fourier mode $A=a(s,t)e^{im[\varphi -\Omega (s)t]}$ therefore obeys the equation in the title. Indeed, each radial derivative acting on the exponential replaces ${\partial }_s$ by ${\partial }_s-imt{\Omega }^{\prime }$, while the azimuthal Laplacian contributes $-m^2/s^2$. At zero magnetic diffusivity, $a$ is constant in time, but the physical magnetic field can still be amplified algebraically by differential rotation.