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

The long-time asymptotic behavior of the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric nonzero boundary conditions at infinity is characterized by using the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. First, the IST is formulated over a single sheet of the complex plane without introducing a uniformization variable. The solution of the focusing NLS equation with nonzero boundary conditions is thus associated with a suitable matrix Riemann-Hilbert problem whose jumps grow exponentially with time for certain portions of the continuous spectrum. This growth is the signature of the well-known modulational instability within the context of the IST. This growth is then removed by suitable deformations of the Riemann-Hilbert problem in the complex spectral plane. Asymptotically in time, the $xt$-plane is found to decompose into two types of regions: a left far-field region and a right far-field region, where the solution equals the condition at infinity to leading order up to a phase shift, and a central region in which the asymptotic behavior is described by slowly modulated periodic oscillations. In the latter region, it is also shown that the modulus of the leading order solution, which is initially obtained in the form of a ratio of Jacobi theta functions, eventually reduces to the well-known elliptic solution of the focusing NLS equation. These results provide the first characterization of the long-time behavior of generic perturbations of a constant background in a modulationally unstable medium.