Cubic Stokes dispersion symmetry (${\nu }_{1,2}=(-k\pm \sqrt{4-3k^2})/2$)

For the linear dispersive Stokes equation, $\omega (\nu )=\omega (k)$ factors as $(\nu -k)({\nu }^2+k\nu +k^2-1)=0$. In $D_{+}=\{\mathrm{Im}k>0,\mathrm{Re}\omega (k)<0\}$ the two displayed other roots lie in the lower half-plane, so both can be used in the global relation. To check the root locations, at $k=ib$ they have imaginary part $-b/2$. A root could reach the real axis only where $\mathrm{Re}\omega (k)=0$, so continuity preserves their signs throughout this connected domain. The branch points $\pm 2/\sqrt{3}$ lie outside its closure. The weights $A_1=(k-{\nu }_2)/({\nu }_1-{\nu }_2)$ and $A_2=({\nu }_1-k)/({\nu }_1-{\nu }_2)$ obey $A_1+A_2=1$ and ${\nu }_j^{\prime }=-A_j$.