Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 61 / 1

Let $E=e^{-ikx+w(k)t}$. Expanding the proposed local relation gives $E[q_t+wq+X_x-ikX]$. Choose the derivative terms in $X$ to cancel the mixed spectral terms:Indeed,so the resulting divergence is exactly $E(q_t+q_x-q_{xxx})$. This establishes the identity for every complex spectral parameter $k$, with the given differentiability. The cubic dispersion relation has the reverse sign from the usual $q_t+q_{xxx}=0$ convention; that sign affects both the spectral sectors and the number of required boundary conditions.

Use the Half-range Fourier transform and the finite-time spectral boundary transformswhere $g_j={\partial }_x^{j}q(0,\cdot )$ for $j=0,1,2$. The spatial transforms are holomorphic in $\mathrm{Im}k<0$, while these finite-time transforms are entire functions. For now $g_2$ is an unknown boundary trace. Integrating the local relation over the half-strip, using spatial decay at infinity, gives the global relationThe boundary sign follows from integrating the flux derivative: its contribution is minus the flux at zero.

Fourier inversion first gives the initial real-line integral plus a real-line integral of $F$. To put the boundary contribution on its useful complex contour, defineOrient its full boundary with $D_{+}$ on the left. It consists of the real axis from left to right and the upper curve $C:k=u+i\sqrt{1+3u^2}$ from right to left. The real-axis piece must not be discarded for this dispersion. The contour integration representation isTo justify the replacement, the integral of $e^{ikx-wt}F$ along $C$ is zero by closing above $C$. In that region $\mathrm{Re}w\ge 0$, and each boundary term is an integral of $e^{-w(t-s)}$ over $0\le s\le t$ times a polynomial in $k$. This has no growing temporal exponential there; $e^{ikx}$ suppresses the upper closing arc for $x>0$. The Cauchy integral theorem then applies. The figure shows the full contour and, separately, the sectors used in the nonlinear problem.

One may replace $F(k,t)$ by $F(k,T)$ for any finite $T\ge t$. Their difference is a future-time integral over $t<s<T$; its factor $e^{w(s-t)}$ decays inside $D_{+}$, so its integral around $\partial D_{+}$ vanishes. All contour formulas are understood as limits of truncated contours, with standard small displacements if a boundary transform needs an Abel limit. Compatibility of the initial and boundary data controls the corner behavior. This is the two-trace half-line cubic dispersion representation.

The other roots of $w(\nu )=w(k)$ satisfyFor $k\in D_{+}$, select the unique other root $\nu (k)$ with negative imaginary part. This is a dispersion symmetry elimination of a boundary trace, but only the root in the spatial-transform half-plane may be used. The root is holomorphic throughout $D_{+}$: the upper branch point $2i/\sqrt{3}$ lies above its boundary. Near the real axis one other root is above and one below; a root can cross the real axis only when $\mathrm{Re}w=0$, so the lower-root choice persists throughout $D_{+}$.

Evaluate the global relation at $\nu$, using the same time transforms because $w(\nu )=w(k)$:Consequentlywhere the completely known spectral density isThe remaining term in the solution formula has integralIts transformed argument stays in the lower half-plane, it is holomorphic inside $D_{+}$, and $e^{w(T-t)}$ does not grow there. Spatial exponential decay closes the truncated domain. Thus the formula in part (b), with $F$ replaced by $H$, uses only the prescribed initial value and the two prescribed boundary traces. It contains no unknown $g_2$ or final-time solution transform. Keeping the inadmissible upper root instead would invalidate this elimination.

For asymptotic analysis it is useful to separate the spectral density from the observation time. For a fixed horizon $T\ge t$, $H(k,T)$ already has this property. If the boundary traces extend to all positive times with suitable growth, their one-sided Laplace transforms give the infinite-horizon versionFor bounded traces these integrals converge in $D_{+}$; boundary values are taken by an Abel limit where needed. The finite-horizon formula permits this replacement: the tail from $t$ to infinity contributes an analytic, decaying integrand with zero integral around $\partial D_{+}$. The resulting spectral functions are independent of $t$.

They can be made particularly explicit on the two contour pieces. On the real axis and upper curve, respectively,The second identity follows because $k^3+k$ is real on $C$: the remaining roots are $\overline{k}$ and $-2\mathrm{Re}k$. Therefore rewrite the solution asThis isolates all large-time dependence in the exponential, allowing the stationary phase method or the method of steepest descent. Along $x/t=\xi$, the phase is $i[k\xi -k-k^3]$, and its stationary points satisfy $3k^2=\xi -1$. Thus the real saddles coalesce at $\xi =1$; any poles arising from the boundary Laplace transforms also have to be retained during contour deformation.

The source specifies smooth boundary traces, not their large-time decay or growth. Accordingly a universal numerical long-time profile cannot be inferred. The infinite-horizon formula requires the stated additional transform hypothesis; for more general data the finite-horizon density $H(k,T)$ remains valid, or an appropriately shifted Laplace contour must be used. This qualification does not leave an unknown boundary trace in the representation.