= Solution
The <dispersion relation> has two additional arguments with the same value. Indeed,
$$
\omega(\nu)=\omega(k)\quad\Longleftrightarrow\quad
(\nu-k)(\nu^2+k\nu+k^2-1)=0.
$$
Thus use the <cubic Stokes dispersion symmetry>
$$
\nu_{1,2}(k)=\frac{-k\pm\sqrt{4-3k^2}}2,
\qquad \nu_1+\nu_2=-k,\quad\nu_1\nu_2=k^2-1.
$$
Both roots are in the lower half-plane for $k\in D_+$. At $k=iv$, $v>0$, their imaginary parts are $-v/2$. A root cannot cross the real axis in $D_+$, since a real root would make $\operatorname{Re}\omega(k)=\operatorname{Re}\omega(\nu)=0$. Continuity therefore proves the claim throughout this connected domain. The branch points $\pm2/\sqrt3$ are outside $\overline{D_+}$; interchanging the two root labels does not change the formula below.
Evaluate the <global relation> at each $\nu_j$, where its analyticity requirement holds:
$$
i\nu_jG_1+G_2=e^{\omega t}\widehat q(\nu_j,t)-\widehat q_0(\nu_j)-(1-\nu_j^2)G_0.
$$
The left side is linear in $\nu_j$. Interpolate it to the argument $k$ with weights
$$
A_1(k)=\frac{k-\nu_2}{\nu_1-\nu_2},\qquad
A_2(k)=\frac{\nu_1-k}{\nu_1-\nu_2}.
$$
They satisfy $A_1+A_2=1$ and $A_1\nu_1+A_2\nu_2=k$. Hence
$$
B(k,t)=e^{\omega t}\sum_{j=1}^2 A_j\widehat q(\nu_j,t)
-\sum_{j=1}^2A_j\widehat q_0(\nu_j)+(1-3k^2)G_0.
$$
For the last coefficient, use
$$
A_1\nu_1^2+A_2\nu_2^2=k(\nu_1+\nu_2)-\nu_1\nu_2=1-2k^2;
$$
adding $1-k^2$ and subtracting the interpolated $1-\nu_j^2$ gives $1-3k^2$.
The remaining solution-transform contribution is $\int_{\partial D_+}e^{ikx}\sum_jA_j\widehat q(\nu_j,t)\,dk$. Its <integrand> is <holomorphic> in $D_+$: the transformed arguments remain in the lower half-plane. The weights are bounded at infinity, and $e^{ikx}$ supplies decay on the upper closing arc for $x>0$. The <Cauchy integral theorem> therefore makes this <integral> zero. Put
$$
F(k)=A_1(k)\widehat q_0(\nu_1(k))+A_2(k)\widehat q_0(\nu_2(k)).
$$
The <Dirichlet spectral representation for the linear dispersive Stokes equation> is now
$$
\boxed{q(x,t)=\frac1{2\pi}\int_{\mathbb R}e^{ikx-\omega(k)t}\widehat q_0(k)\,dk
+\frac1{2\pi}\int_{\partial D_+}e^{ikx-\omega(k)t}\bigl[(1-3k^2)G_0(k,t)-F(k)\bigr]dk.}
$$
All quantities in this expression are transforms of the prescribed initial and boundary values. Neither derivative trace remains.
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