Long-Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

Abstract

We characterize the long-time asymptotic behavior of the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity by using a variant of 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, we formulate the IST over a single sheet of the complex plane without introducing the uniformization variable that was used by Biondini and Kovačič in 2014. The solution of the focusing NLS equation with nonzero boundary conditions is thereby associated with a 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. We then eliminate this growth by performing suitable deformations of the Riemann-Hilbert problem in the complex spectral plane. The results demonstrate that the solution of the focusing NLS equation with nonzero boundary conditions remains bounded at all times. Moreover, we show that, asymptotically in time, the xt-plane decomposes 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. Finally, we show how, in the latter region, the modulus of the leading-order solution, initially obtained as a ratio of Jacobi theta functions, can be reduced to the well-known elliptic solutions 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. © 2017 Wiley Periodicals, Inc.