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In a recent paper, we presented scenarios of long-time asymptotics for asolution of the focusing nonlinear Schr\"odinger equation whose initial dataapproach two different plane waves$A_j\mathrm{e}^{\mathrm{i}\phi_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ atminus and plus infinity. In the shock case $B_1<B_2$ some scenarios includesectors of genus $3$, that is sectors $\xi_1<\xi<\xi_2$, $\xi:=\frac{x}{t}$where the leading term of the asymptotics is expressed in terms ofhyperelliptic functions attached to a Riemann surface $M(\xi)$ of genus $3$.The long-time asymptotic analysis in such a sector is performed in anotherrecent paper. The present paper deals with the asymptotic analysis in atransition zone between two genus $3$ sectors $\xi_1<\xi<\xi_0$ and$\xi_0<\xi<\xi_2$. The leading term is expressed in terms of elliptic functionsattached to a Riemann surface $\tilde{M}$ of genus $1$. A central step in thederivation is the construction of a local parametrix in a neighborhood of twomerging branch points. We construct this parametrix by solving a model problemwhich is similar to the Riemann-Hilbert problem associated with the Painlev\'eIV equation.

The focusing NLS equation with step-like oscillating background: asymptotics in a transition zone