Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 61 / 1

Set $\kappa =(a^2/3-b{)}^{1/2}>0$, $\sigma =-c+ab/3-2a^3/27$, and use the gauge reduction of a third-order half-line equationThe spatial differential operator acting on $q$ is obtained by replacing ${\partial }_X$ with $\kappa {\partial }_x-a/3$. Its second-derivative coefficient vanishes. The first-derivative coefficient is $\kappa (b-a^2/3)=-{\kappa }^3$, and the constant coefficient is $c-ab/3+2a^3/27=-\sigma$. The time derivative supplies $\sigma q+{\kappa }^3{q}_t$, so all constant terms cancel and the remaining equation is ${\kappa }^3(q_t+q_{xxx}-q_x)=0$.

The transformed initial condition and Dirichlet boundary data are thereforeCompatibility at the corner is preserved. Since $a<0$, this exponential gauge improves the spatial decay of the initial data, which supports convergence of its Half-range Fourier transform.

Put $w(k)=-i(k^3+k)$ and define the spectral fluxWriting $E=e^{-ikx+w(k)t}$, division of the proposed divergence form by $E$ gives $q_t+wq+X_x-ikX$. Direct differentiation yieldsThe last term cancels $wq$, so the divergence form is exactly the normalized partial differential equation. This calculation fixes both the sign of the dispersion relation and the sign of the boundary flux used below.

Use the negative-exponential Half-range Fourier transformFor the boundary traces $g_j(t)={\partial }_x^{j}q(0,t)$, define finite-time spectral boundary transforms $G_j(k,t)={\int }_0^t{e}^{w(k)s}{g}_j(s)ds$. Integrating the conservation law in $x$ and then time gives the global relationThe plus sign of the boundary contribution follows because the integrated flux is $-E(0,t)X(0,t,k)$.

Fourier inversion consequently gives a real-line initial integral plus a boundary integral. To put the latter in the complex plane, defineOrient $\partial D_{+}$ from the left upper infinity, through $i$, to the right upper infinity, with $D_{+}$ on the left. Each $G_j$ is entire, and in the region between the real axis and this contour, $\mathrm{Re}w\ge 0$. The causal factors $e^{-w(t-s)}$ are bounded there and $e^{ikx}$ decays for $x>0$. The Cauchy integral theorem, with the usual large-arc limiting argument, deforms the boundary integral to giveInitial smoothness, spatial decay and corner compatibility justify inversion; if necessary the integrals are first regularized before contour deformation. The unknown Neumann-type traces $g_1,g_2$ will be removed by the global relation.

The cubic dispersion symmetry with negative transport supplies two other roots of $w(\nu )=w(k)$:Both lie in the lower half-plane when $k\in D_{+}$. For example this is immediate at large positive imaginary $k$. A root could leave that half-plane only by becoming real, where $\mathrm{Re}w(\nu )=0$, contradicting $\mathrm{Re}w(k)<0$. Connectedness of $D_{+}$ completes the argument; a coalescence merely exchanges the two roots.

Evaluate the global relation at these roots. Because their dispersion values equal $w(k)$, the same boundary transforms occur, andInterpolate this linear polynomial in $\nu$ at ${\nu }_1,{\nu }_2$. Defineand similarly $\mathcal{T}\hat{q}(k,t)$. Substitution givesIndeed, interpolation of ${\nu }^2+1$ yields $({\nu }_1+{\nu }_2)k-{\nu }_1{\nu }_2+1=-2k^2$, which proves the coefficient $-(3k^2+1)$ after subtracting $k^2+1$.

The final-state term has zero contour integral. After cancelling $e^{wt}$ its integrand is $e^{ikx}\mathcal{T}\hat{q}(k,t)$, holomorphic in $D_{+}$ and decaying upon closing the contour upward for $x>0$. The possible square-root singularities are removable: the expression is unchanged by exchanging the roots, and when they coalesce at $\nu$, its limit is $F(\nu )+(k-\nu )F^{\prime }(\nu )$, with the same formula for the final-state transform. Thus the Cauchy integral theorem eliminates this unknown term.

We obtain the fully specified Dirichlet spectral formula for Airy flow with negative transportOnly the initial Fourier transform and the given boundary-value time transform remain.

First remove the observation time from the spectral amplitude. For any fixed horizon $H>t$, replace $G_0(k,t)$ by $G_0(k,H)$ in the formula of part (d). The difference in the boundary integral is proportional toFor every $s>t$, this integrand is holomorphic inside $D_{+}$, where $\mathrm{Re}w<0$, and decays on its large closing arcs. The Cauchy integral theorem makes the difference zero. This finite-horizon extension of a boundary transform therefore gives the time-independent spectraAll observation-time dependence is now in $e^{ikx-wt}$.

When the boundary history is sufficiently decaying, the natural long-time version isThe integral converges in $\mathrm{Re}w<0$ for bounded boundary data and on its boundary for integrable histories; otherwise one must state an Abel prescription or a suitable temporal growth assumption. Arbitrary smooth $g_0$ need not have such an infinite-time transform. The finite-horizon formula remains exact without that extra assumption, and the source alone does not prescribe a universal long-time limit.

A useful further decomposition, if $g_0(s)=g_{\infty }+r(s)$ with integrable $r$, isThis separates the persistent forcing from the transient spectrum. The pole at $k=i$, where $w=0$, accounts for the decaying stationary profile $g_{\infty }{e}^{-x}$; that profile also follows directly from $q_{xxx}-q_x=0$, spatial decay and its boundary value. For the remaining oscillatory integrals the phase is $i[kx+(k^3+k)t]$. At a fixed ray $x/t=\xi$, its stationary points satisfy $3k^2+1+\xi =0$, or $k=\pm i\sqrt{(1+\xi )/3}$. These time-independent amplitudes expose precisely the poles and saddle points used in the method of steepest descent; allowable deformations must also respect the analyticity strip of the initial transform.

The algebraic change of variables still works for $a>0$, but the transformed initial data are now multiplied by $e^{aX/3}$. Mere decay of $Q_0$ does not ensure existence of the Half-range Fourier transform used above. For example, $Q_0(X)=e^{-dX}$ with $0<d<a/3$ gives a growing transformed profile.

There is also a genuine stability distinction. On the whole line a real-frequency mode of the original equation evolves with factorIts modulus is $e^{(a{K}^2-c)T}$, so positive $a$ amplifies arbitrarily high frequencies rather than damping them. Thus the formal gauge equivalence does not justify the same unweighted Sobolev well-posedness conclusion. Weighted or more strongly decaying data and corresponding contour changes may allow particular solutions; the assumptions stated for the negative-$a$ construction cannot simply be reused.