  We consider the Cauchy problem for the focusing nonlinear Schr\"odinger
equation with initial data approaching two different plane waves
$A_j\mathrm{e}^{\mathrm{i}\phi_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as
$x\to\pm\infty$. Using Riemann-Hilbert techniques and Deift-Zhou steepest
descent arguments, we study the long-time asymptotics of the solution. We
detect that each of the cases $B_1<B_2$, $B_1>B_2$, and $B_1=B_2$ deserves a
separate analysis. Focusing mainly on the first case, the so-called shock case,
we show that there is a wide range of possible asymptotic scenarios. We also
propose a method for rigorously establishing the existence of certain
higher-genus asymptotic sectors.
