= Solution
Every term in $\widetilde S$ is proportional to $N$. The <Large-N expansion> therefore suppresses fluctuations of the auxiliary field by powers of $1/N$, and the leading partition function comes from a <saddle-point approximation>.
Vary with respect to $i\lambda$ and use $\delta\operatorname{Tr}\log K=\operatorname{Tr}(K^{-1}\delta K)$. At a translation-invariant saddle $i\lambda=m^2$, the <gap equation> is
$$
\boxed{
\frac1{vg}
=T\sum_{n\in\mathbb Z}
\int\frac{d^2k}{(2\pi)^2}
\frac1{\omega_n^2+v^2k^2+m^2},
\qquad \omega_n=2\pi nT.}
$$
The frequencies are the bosonic <Matsubara frequencies> imposed by periodicity around the imaginary-time thermal circle.
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