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

In a recent paper, we presented scenarios of long-time asymptotics for a solution of the focusing nonlinear Schr\"odinger equation whose initial data approach two different plane waves $A_j\mathrm{e}^{\mathrm{i}\phi_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ at minus and plus infinity. In the shock case $B_1<B_2$ some scenarios include sectors 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 of hyperelliptic functions attached to a Riemann surface $M(\xi)$ of genus $3$. The long-time asymptotic analysis in such a sector is performed in another recent paper. The present paper deals with the asymptotic analysis in a transition 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 functions attached to a Riemann surface $\tilde{M}$ of genus $1$. A central step in the derivation is the construction of a local parametrix in a neighborhood of two merging branch points. We construct this parametrix by solving a model problem which is similar to the Riemann-Hilbert problem associated with the Painlev\'e IV equation.