= Solution
The <cubic dispersion symmetry with negative transport> supplies two other roots of $w(\nu)=w(k)$:
$$
\nu_{1,2}(k)=\frac{-k\pm\sqrt{-3k^2-4}}2,\qquad \nu_1+\nu_2=-k,\qquad\nu_1\nu_2=k^2+1.
$$
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 $\operatorname{Re}w(\nu)=0$, contradicting $\operatorname{Re}w(k)<0$. 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, and
$$
G_2+i\nu_jG_1=(\nu_j^2+1)G_0-F(\nu_j)+e^{wt}\widehat q(\nu_j,t),\qquad j=1,2.
$$
Interpolate this linear polynomial in $\nu$ at $\nu_1,\nu_2$. Define
$$
\mathcal TF(k)=\frac{(k-\nu_2)F(\nu_1)+(\nu_1-k)F(\nu_2)}{\nu_1-\nu_2},
$$
and similarly $\mathcal T\widehat q(k,t)$. Substitution gives
$$
G_2+ikG_1-(k^2+1)G_0=-\mathcal TF(k)-(3k^2+1)G_0(k,t)+e^{wt}\mathcal T\widehat q(k,t).
$$
Indeed, 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 $e^{ikx}\mathcal T\widehat q(k,t)$, 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 $F(\nu)+(k-\nu)F'(\nu)$, 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 transport>
$$
\boxed{q(x,t)=\frac1{2\pi}\int_{\mathbb R}e^{ikx-wt}F(k)\,dk-\frac1{2\pi}\int_{\partial D_+}e^{ikx-wt}\bigl[\mathcal TF(k)+(3k^2+1)G_0(k,t)\bigr]dk.}
$$
Only the initial <Fourier transform> and the given boundary-value time transform remain.
Back to article page
