= Solution
Put $A=\gamma-i\omega$ and $B=\alpha-i\omega$. For $\widehat v=0$,
$$
\widehat u=-\frac{ik}{A}\widehat\phi,
\qquad
\widehat\phi=\phi_0\exp\left(\frac{i\beta k}{2A}y^2\right).
$$
Continuity gives the damped <Equatorial Kelvin wave> dispersion relation
$$
\boxed{(\gamma-i\omega)(\alpha-i\omega)+c^2k^2=0.}
$$
Only roots satisfying $\operatorname{Re}(i\beta k/A)<0$ are trapped; for weak damping this is the eastward branch $k\operatorname{Re}\omega>0$. The other algebraic root makes the Gaussian grow with $|y|$ and must be rejected.
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