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A spatially nonuniform system is globally unstable when a normal mode satisfying its global spatial boundary conditions has positive temporal growth rate. This differs from purely local temporal amplification or convective wave-packet instability. A locally absolutely unstable region can be necessary without being sufficient for a growing global mode of finite spatial extent.

Near a spatially extended Hopf bifurcation, a critical oscillatory normal mode has a slowly varying complex amplitude. For a selected travelling-wave branch, a method of multiple scales expansion takes the formHere the distance from onset is of order ${\epsilon }^2$, $v_0$ is the critical eigenvector, and $v_g$ is the group velocity. The comoving coordinate removes the first spatial derivative of the envelope. Projection at the first resonant order gives $A_T=\rho A+D_0{A}_{XX}-G_0|A{|}^{2}A$. Spatial translation and the temporal phase of the Hopf oscillation give a constant-phase symmetry $A\mapsto e^{i\theta }A$, which permits the cubic term $|A{|}^{2}A$ but forbids a generic $A^2$ term. The coefficients are complex because the envelope has both growth and frequency detuning. For a supercritical branch with positive real diffusion, real rescalings and a uniform phase rotation give the complex Ginzburg–Landau equationA subcritical Hopf branch requires higher saturation terms instead. If spatial reflection makes opposite travelling waves critical together, two coupled amplitude equations are required initially; the scalar equation applies to a selected single-wave branch when its competing amplitude is stable.

Distant boundaries can still determine which travelling pattern is observed. They can select its phase and wavenumber, reflect a travelling disturbance, inject a competing wave, or let a growing wave packet leave the domain. In particular, amplification in a comoving frame can be convective wave-packet instability rather than absolute wave-packet instability at a fixed laboratory point. A finite-domain outcome then need not reflect intrinsic bulk modulation dynamics. Periodic boundary conditions are the appropriate idealization when studying bulk instabilities without end selection or reflected-wave forcing: they eliminate physical end layers and make the allowed envelope Fourier modes discrete. They do not assert that real distant boundaries are always negligible. Choose a periodic length compatible with the carrier, fix the phase winding, and compare perturbations within that same periodic problem. On an open physical domain the group velocity and actual boundary conditions must be restored.

The optimum carrier wavenumber corresponds to the spatially uniform envelope $A_f=e^{-icT}$, often called a flat state. Its phase is arbitrary. Write $A=e^{-icT}(1+r)e^{i\varphi }$ with real small amplitude and phase disturbances. To first order,A Fourier mode $e^{\lambda T+ikX}$ therefore has matrix and dispersion relationThe trace is negative. The amplitude mode at $k=0$ has eigenvalue $-2$, while the phase mode is neutral because constant phase shifts preserve the solution. Put $D=1+bc$. For every nonzero real $k$, the determinant is positive if $D\ge 0$, whereas $D<0$ makes it negative for $0<k^2<-2D/(1+b^2)$. ThusThis is the Benjamin-Feir stability condition. At $D=0$, nonzero modes still decay, but the leading long-wave decay is fourth order rather than diffusive. On a periodic envelope interval of length $L$, the smallest nonzero wavenumber is $2\pi /L$, so strict stability of all allowed nonzero modes requiresA sufficiently short periodic domain can therefore exclude the unstable band even when the infinite-domain Benjamin-Feir stability condition fails.

The amplitude is damped while the phase is slow. For a slowly varying phase, the leading slaved amplitude is $r=-\frac{1}{2}\{b{\varphi }_{XX}+({\varphi }_X{)}^2\}+\cdots$. Substitution in the phase equation gives the diffusion term $D{\varphi }_{XX}$ and the nonlinear frequency shift $(c-b)({\varphi }_X{)}^2$. The latter also follows directly from a constant phase gradient: an envelope with gradient $q$ has amplitude squared $1-q^2$ and frequency $c+(b-c)q^2$. Constant phase symmetry rules out undifferentiated $\varphi$; translation invariance gives constant coefficients. The leading comoving Ginzburg-Landau equation also has spatial parity, so odd linear derivatives are absent at this order. Its neutral phase eigenvalue expands asThis coefficient follows by inserting a power series for $\lambda$ in the quadratic dispersion relation. Close to $D=0$, it becomes $E=(1+b^2)/2>0$ at leading order. The long-wave phase reduction of the complex Ginzburg-Landau equation is consequentlyThe fourth derivative regularizes negative phase diffusion. Terms omitted here have higher order in the long-wave expansion. First-order advection has been removed by the group-velocity frame; higher odd derivatives in a more general travelling-wave problem are beyond this leading amplitude approximation.

Take $D=-d$ with $d>0$ small. At the instability boundary $c=-1/b$, so $g=-(1+b^2)/b$ is nonzero. Set $\xi =\sqrt{d/E}X$, $\tau =d^{2}T/E$, and $\varphi =(d/g)\Phi$. At leading order the phase dynamics become the potential Kuramoto-Sivashinsky equation,With $u=-2{\Phi }_{\xi }$ this isThe variable $u$ is a phase-gradient disturbance, not the original physical field. Periodicity of the phase fixes ${\int }_0^{P}ud\xi =0$ for the zero-winding branch; the Kuramoto-Sivashinsky equation conserves this mean. The scaled period is $P=L\sqrt{d/E}$.

At fixed period $P$, the flat phase has growth rates ${\lambda }_n=k_n^2-k_n^4$, $k_n=2\pi n/P$. Its first loss of stability occurs at $P=2\pi$. To find what replaces it, put $k=2\pi /P$ and consider $P$ just above $2\pi$. After choosing an origin by translation, expand a small steady modulation as $u=a\sin (k\xi )+b_2\sin (2k\xi )+\cdots$. Projection of $-u{u}_{\xi }$ onto the first two harmonics yieldsSince ${\lambda }_2=-12$ at onset, this harmonic is slaved to $b_2=k{a}^2/(2{\lambda }_2)+\cdots$. The fundamental amplitude obeysThe primary periodic bifurcation of the Kuramoto-Sivashinsky equation is therefore supercritical. For $0<P-2\pi \ll 1$,The radial eigenvalue of this nonzero branch is $-2{\lambda }_1+O({\lambda }_1^2)<0$; all higher harmonics remain damped and the modulation's spatial phase is neutral. Thus the small finite-amplitude modulation is stable in the sense of orbital stability, at fixed mean and fixed period, despite the flat state's instability. The two signs of $a$ represent translations of the same modulation. In the potential formulation the spatial mean of $\Phi$ advances at $⟨({\Phi }_{\xi }{)}^2⟩$: the original wave acquires a nonlinear frequency correction rather than requiring a time-independent absolute phase.

For larger $P$, further linear modes become active and the small-amplitude argument no longer determines stability. A periodic modulation $\bar{u}$ must then be tested with the operator $v_{\tau }=-v_{\xi \xi }-v_{\xi \xi \xi \xi }-{\partial }_{\xi }(\bar{u}v)$ on mean-zero periodic disturbances. Its translation mode is always neutral, and secondary oscillatory or more complicated modulation can occur. Moreover, stability against perturbations of the same period does not prove stability against disturbances of a larger period; those require a separate sideband or Floquet theory calculation. The controlled conclusion near the first threshold is the existence of a stable small modulation family, not universal stability of all periodic Kuramoto-Sivashinsky solutions.

The Briggs-Bers technique starts with a temporal Laplace transform and spatial Fourier transform of a localized disturbance. The temporal inversion contour must initially lie above every temporal singularity. Lower it while deforming the spatial contour away from the roots of the dispersion relation. A collision of spatial branches originating on opposite sides of that contour can create a spatial pinch point. A contributing pinch with $\mathrm{Im}{\omega }_{\ast }>0$ gives absolute wave-packet instability: exponential growth at a fixed position. Growth that is transported away while a fixed position decays is convective wave-packet instability. Solving $D=D_k=0$ finds candidate double roots, but does not by itself establish a pinch.

Substitution of the normal mode givesFor real wavenumber $k$, the temporal growth rate isA nonzero real cubic coefficient in this expression makes it unbounded above on one end of the real axis. Once that coefficient vanishes, a positive quadratic coefficient also makes it unbounded above. Hence the bounded temporal growth condition for cubic dispersion isThe maximum is then $-\mathrm{Re}\gamma$, attained at $k=0$ and, when $\mathrm{Re}\alpha =0$, at all real $k$. A finite upper bound is necessary for choosing the initial temporal inversion contour; arbitrarily fast high-wavenumber growth prevents the usual Briggs-Bers criterion from defining that causal inversion for generic localized data.

For $\beta \ne 0$, the stationary candidates satisfysoEach candidate must pass the spatial-branch pinch test before its imaginary frequency is used. In particular, under the finite-growth conditions the real-wavenumber evolution has multiplier modulus at most $e^{-t\mathrm{Re}\gamma }$. For integrable transformed initial data this bounds its fixed-position amplitude by a constant times that exponential. Thus a further necessary condition isThe false complex saddle in dissipative cubic dispersion illustrates why the second formal candidate cannot override this bound. For $\mathrm{Re}\alpha <0$, the neighborhood $k=0$ gives a fixed-position impulse response proportional to $e^{-\gamma t}/\sqrt{t}$: the cubic term is smaller on its $k=O(t^{-1/2})$ scale. It therefore supplies the usual growing pinch when $\mathrm{Re}\gamma <0$. The purely dispersive cases have stationary-phase algebraic prefactors instead. If $\beta =0$, only the quadratic stationary point remains; if both derivative coefficients vanish, the equation is local multiplication and its growth is directly $e^{-\gamma t}$.

Use normal modes $e^{ikx+st}$, so $s=-i\omega$ in the convention $e^{ikx-i\omega t}$. For a translation-invariant, well-posed linear initial value problem on the whole line, localized data are propagated by a causal Green function. The Fourier transform in space and Laplace transform in time give an inverse integral whose denominator is the dispersion relation $D(k,s)$. The initial temporal contour is a Bromwich line to the right of every real-wave-number temporal singularity.

The Briggs-Bers technique lowers that temporal contour while deforming the spatial contour to avoid its poles. A spatial pinch point occurs when two spatial branches originating on opposite sides of the original Fourier contour collide and prevent further deformation. Algebraically a simple candidate satisfies $D=D_k=0$, with $D_s\ne 0$ and $D_{kk}\ne 0$; these equations alone do not establish the pinch topology. A same-side collision is a false spatial saddle in quartic dispersion. The opposite-side requirement is also explicit in the primary numerical study of pinch-point detection.

The rightmost contributing pinch determines the fixed-position exponential growth of the impulse response, ordinarily with a $t^{-1/2}$ saddle point prefactor. A positive rate is absolute wave-packet instability. If real-wave-number normal modes grow but the impulse response decays at each fixed position, it is convective wave-packet instability; growth can then occur along moving rays. If no temporal normal mode grows, the system is stable, with equality thresholds treated as marginal rather than exponential decay. Along a ray $x=vt$, apply the same argument to $s(k)+ikv$, whose saddle point condition is $s^{\prime }(k)+iv=0$. Thus the laboratory frame is part of the absolute/convective definition.

For the quartic problem the temporal dispersion relation isWrite $a=\mathrm{Re}{a}_1$, $b=\mathrm{Re}{a}_2$, $c=\mathrm{Re}{a}_3$. For $b>0$ the causal Fourier kernel is absolutely convergent for positive time and the finite maximum temporal growth rate isThe usual initial-value framework fails if $b<0$, or if $b=0$ and $a>0$, because arbitrarily large real wave numbers grow arbitrarily fast: no finite Bromwich line can start to their right. This is a high-frequency ill-posedness obstruction, not an absolute instability criterion. If $b=0$ and $a\le 0$, the growth bound is finite, but the dispersive limiting case needs oscillatory Fourier inversion or a limiting-absorption construction when there is no quadratic damping. It must not be justified by the absolutely convergent quartic-diffusion argument. The following explicit calculation first assumes $b>0$ and $a_2\ne 0$.

The double-root candidates areRather than assume all are physical, compute the causal resolvent at the impulse location. Put $S=s+a_3$, and for large $\mathrm{Re}s$ factorThen ${\kappa }_1{\kappa }_2=\sqrt{S/a_2}$ and $({\kappa }_1+{\kappa }_2{)}^2=-a_1/a_2+2\sqrt{S/a_2}$, with sheets selected from that large-$s$ region. Partial fraction decomposition and ${\int }_{\mathbb{R}}(k^2+{\kappa }^2{)}^{-1}dk=\pi /\kappa$ give the quartic impulse-response Laplace resolventAll roots in this expression are continued from the causal region, not selected afresh at a candidate collision. The two possible singularities are now transparent. $S=0$ is a square-root singularity when $a_1\ne 0$. The extra singularity requires $\sqrt{S}=a_1/(2\sqrt{a_2})$. Taking $\sqrt{a_2}$ with positive real part and putting $z=a_1/\sqrt{a_2}$, this lies on the physical $\sqrt{S}$ sheet precisely when $\mathrm{Re}z>0$. If that real part is zero the candidate is on a cut boundary and has nonpositive real $S$, so it cannot lie to the right of $S=0$. For $\mathrm{Re}z<0$, the apparently growing candidate may be on the wrong sheet altogether.

Therefore the physical-sheet growth rate of a quartic impulse response isThese are resolvent branch singularities on the causal continuation, so the formula implements the pinch test instead of merely listing formal saddle point rates. For $a_1=0$, the zero saddle point is fourth-order and the resolvent is proportional to $S^{-3/4}$; its impulse response is proportional to $t^{-1/4}{e}^{-a_{3}t}$ and the same exponential-rate formula applies. At nondegenerate candidates the prefactor is $t^{-1/2}$.

The classification in this diffusive case is now explicit: ${\sigma }_T<0$ gives stability; ${\sigma }_A>0$ gives absolute instability; ${\sigma }_T>0$ with ${\sigma }_A<0$ gives convective instability. When ${\sigma }_T>0$ and ${\sigma }_A=0$, the nondegenerate fixed-position response still decays algebraically, at the marginal absolute/convective threshold. Other equality cases require the stated saddle point prefactors. Temporal instability supplies a growing ray: at a real maximum of $\mathrm{Re}s(k)$, choose $v=-\mathrm{Im}{s}^{\prime }(k)$ so that the moving-ray saddle point is stationary there.

Three examples show why complex coefficients and sheets matter. For $(a_1,a_2,a_3)=(2,1,1/2)$, ${\sigma }_T={\sigma }_A=1/2$, so the instability is absolute. For $(1+2i,1,1/10)$, ${\sigma }_T=3/20>0$ but ${\sigma }_A=-1/10$, so it is convective even though there is no explicit first-derivative advection term: dispersion carries packets in opposite directions. For $(-2,1,1/2)$, both physical rates are $-1/2$. The formal candidates $k=\pm i$ have $s=1/2$ but fail the physical-sheet condition, so this temporally stable example is not absolutely unstable.

On the admissible boundary $b=0$, $a\le 0$, $a_2\ne 0$, the oscillatory or damped Fourier construction has ${\sigma }_T=-c$. The surviving zero-wave-number stationary response has the same exponential rate $-c$, with a quadratic or quartic algebraic factor; damping/dispersion therefore does not generate a separate convective-growth range there. If $a_2=0$, the problem is second order and its diffusion/dispersive kernel must be used instead of the quartic formula. Application to a finite interval or nonlocalized data also changes the spatial inversion problem, so the whole-line localized-data hypothesis is essential.

Take the spatial Fourier transform of the causal Green function. The transformed equation is ${\hat{G}}_t=(\mu -iUk-\gamma k^2)\hat{G}+\delta (t)$, with zero response for $t<0$. Integration across the impulse sets $\hat{G}(k,0^{+})=1$, soInverting the Gaussian Fourier transform givesThis solves the homogeneous equation for $t>0$ and tends to the Dirac delta distribution as $t\downarrow 0$. Its spatial integral is $e^{\mu t}$, as required by the zero-wavenumber growth law.

Along a ray $x=vt$, the exponential growth rate isThe impulse grows in the moving interval of rays $U-2\sqrt{\gamma \mu }<v<U+2\sqrt{\gamma \mu }$, while its peak at $x=Ut$ grows as $t^{-1/2}{e}^{\mu t}$. At a fixed position,Thus, under the printed assumption $\mu >0$, $0<\mu <U^2/(4\gamma )$ gives convective wave-packet instability, because the growing disturbance is swept away from each fixed observation point. $\mu >U^2/(4\gamma )$ gives absolute wave-packet instability, with exponential growth at fixed position. At equality the fixed-position exponential rate is zero and the impulse decays algebraically as $t^{-1/2}$, even though its moving peak grows. The two bounding rays have the same algebraic prefactor.

These statements classify the Green function and localized initial disturbances propagated by its convolution. They do not assert that every possible initial condition decays at fixed position in the convective case: spatially uniform initial data instead produce $e^{\mu t}$. Persistent external forcing can likewise change the observed late-time response. The distinction between convective wave-packet instability and absolute wave-packet instability concerns localized disturbances in a specified frame.

Retain the factor $ϵ$ in the last temporal phase, as printed in the original PDF. Let $a=U/(2\gamma )$ and put $\eta =e^{ax}b(\xi )e^{st}$ with $s=-i{\omega }_0-iϵ{\omega }_1$. The drift-removing substitution for a Dirichlet Sturm-Liouville problem cancels the first derivative:With $\xi =\sqrt{ϵ}fx$, this becomesChoose the positive scale $f=(\nu /\gamma {)}^{1/4}$. To leave an order-one spectral parameter, cancel the numerator's order-one term. ThenThe assumed decaying solutions of the Hermite differential equation require $\lambda =2j+1$, $j=0,1,\dots$. They are proportional to $e^{-{\xi }^2/2}{H}_j(\xi )$, with physicists' Hermite polynomials. Therefore the global modes of a quadratically confined Ginzburg-Landau equation haveFor every fixed positive $ϵ$, multiplying the Hermite-Gaussian by $e^{Ux/(2\gamma )}$ still gives decay at both infinities. The strongest mode is $j=0$, so the whole-line system is globally unstable precisely whenEquality is a neutral ground mode. This spectrum is exact for the stated quadratic profile, even though the notation organizes it as an expansion in $ϵ$.

The homogeneous absolute-instability threshold in part (a) is $\mu =U^2/(4\gamma )$. The global threshold approaches that value as the profile varies increasingly slowly, but at finite $ϵ$ the quadratic confinement adds the loss $ϵ\sqrt{\gamma \nu }$. A locally absolutely unstable patch at the profile maximum is necessary but must be strong and broad enough to overcome that loss. Merely having positive local temporal growth ${\mu }_0>0$, which can be only convective wave-packet instability, is insufficient for a growing decaying global normal mode. This connects the local impulse classification to the discrete global spectrum.

Put $z=a_1/\sqrt{a_2}$ using the root with positive real part. The additional resolvent singularity at $s+a_3=z^2/4$ lies on the physical square-root sheet only when $\mathrm{Re}z>0$; otherwise the apparent saddle is on the other sheet. The singularity at $s=-a_3$ remains, including a quartic degeneracy when $a_1=0$. The rightmost physical singularity determines the exponential fixed-position growth rate. Compare it with the temporal maximum $-\mathrm{Re}{a}_3+\max (\mathrm{Re}{a}_1,0{)}^2/(4\mathrm{Re}{a}_2)$ to distinguish absolute wave-packet instability from convective wave-packet instability. Equality cases require their algebraic prefactors.

The response to a localized perturbation must be followed in space as well as time. Growth at a fixed position is absolute wave-packet instability; growth transported away while the original position decays is convective wave-packet instability. The Briggs-Bers criterion evaluates the relevant complex spatial branches of a dispersion relation, while a real-wavenumber temporal mode test alone does not generally distinguish those behaviors.