Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 68 / 3

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.

Seek an outer asymptotic expansion $y=y_0+ϵy_1+\cdots$ for fixed $x>0$. The leading equation is $x{y}_0^{\prime }=0$, and the endpoint condition gives $y_0=1$. At first order, $x{y}_1^{\prime }=y_0=1$ and $y_1(1)=0$, so $y_1=\log x$. This outer correction cannot be evaluated at $x=0$.

The coefficient $x+ϵ$ selects the boundary layer coordinate $X=x/ϵ$. With $Y(X)=y(ϵX)$, the equation becomes $(1+X)Y_X=ϵY$. Its inner leading term is constant; matching it to the outer solution gives $Y_0=1$. At first order, $Y_{1X}=1/(1+X)$, hence $Y_1=\log (1+X)+C$. In the overlap $1\ll X\ll {ϵ}^{-1}$, the outer solution is $1+ϵ\log ϵ+ϵ\log X$. Matching the constants therefore givesfor fixed $X$. The logarithmic matching in a small-exponent boundary layer must retain $ϵ\log ϵ$, which is larger in magnitude than a plain order-$ϵ$ term. At the endpoint,The coefficient of a separate plain $ϵ$ term is zero.

Direct integration provides the exact check:At zero its logarithm is $ϵ\log ϵ-ϵ\log (1+ϵ)=ϵ\log ϵ-{ϵ}^2+O({ϵ}^3)$. Exponentiating gives $y(0)=1+ϵ\log ϵ+O({ϵ}^2{\log }^2ϵ)$, whose remainder is indeed $o(ϵ)$. The first-order uniform composite $1+ϵ\log [(x+ϵ)/(1+ϵ)]$ also satisfies the boundary condition exactly and reproduces both expansions to the requested order.