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The long-time asymptotic behavior of the focusing nonlinear Schr\"odinger(NLS) equation on the line with symmetric nonzero boundary conditions atinfinity is characterized by using the recently developed inverse scatteringtransform (IST) for such problems and by employing the nonlinear steepestdescent method of Deift and Zhou for oscillatory Riemann-Hilbert problems.First, the IST is formulated over a single sheet of the complex plane withoutintroducing a uniformization variable. The solution of the focusing NLSequation with nonzero boundary conditions is thus associated with a suitablematrix Riemann-Hilbert problem whose jumps grow exponentially with time forcertain portions of the continuous spectrum. This growth is the signature ofthe well-known modulational instability within the context of the IST. Thisgrowth is then removed by suitable deformations of the Riemann-Hilbert problemin the complex spectral plane. Asymptotically in time, the $xt$-plane is foundto decompose into two types of regions: a left far-field region and a rightfar-field region, where the solution equals the condition at infinity toleading order up to a phase shift, and a central region in which the asymptoticbehavior is described by slowly modulated periodic oscillations. In the latterregion, it is also shown that the modulus of the leading order solution, whichis initially obtained in the form of a ratio of Jacobi theta functions,eventually reduces to the well-known elliptic solution of the focusing NLSequation. These results provide the first characterization of the long-timebehavior of generic perturbations of a constant background in a modulationallyunstable medium.

Long-time asymptotics for the focusing nonlinear Schr\"odinger equation  with nonzero boundary conditions at infinity and asymptotic stage of  modulational instability