The initial-boundary value problem (ibvp) for the cubic nonlinear Schrödinger (NLS) equation on the half-line with data in Sobolev spaces is analysed via the formula obtained through the unified transform method, and a contraction mapping approach. First, the linear Schrödinger (LS) ibvp with initial and boundary data in Sobolev spaces is solved and the basic space and time estimates of the solution are derived. Then, the forced LS ibvp is solved for data in Sobolev spaces, on the half line [0, ∞) for the spatial variable and on an interval [0, T ], 0 < T < ∞, for the temporal variable, by decomposing it into a free ibvp and a forced ibvp with zero data, and its solution is estimated appropriately. Furthermore, using these estimates, well-posedness of the NLS ibvp with data (u(x, 0), u(0, t)) in H s x (0, ∞) × H (2s+1)/4 t (0, T), s > 1/2, is established via a contraction mapping argument. In addition, this work places Fokas' unified transform method for evolution equations into the broader Sobolev spaces framework.

We characterize the nonlinear stage of modulational instability (MI) by studying the long-time asymptotics of the focusing nonlinear Schrödinger (NLS) equation on the infinite line with initial conditions tending to constant values at infinity. Asymptotically in time, the spatial domain divides into three regions: a far left and a far right field, in which the solution is approximately equal to its initial value, and a central region in which the solution has oscillatory behavior described by slow modulations of the periodic traveling wave solutions of the focusing NLS equation. These results demonstrate that the asymptotic stage of MI is universal, since the behavior of a large class of perturbations characterized by a continuous spectrum is described by the same asymptotic state.

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.

We characterize the properties of the asymptotic stage of modulational instability arising from localized perturbations of a constant background, including the number and location of the individual peaks in the oscillation region. We show that, for long times, the solution tends to an ensemble of classical (i.e., sech-shaped) solitons of the focusing nonlinear Schrödinger equation (as opposed to the various breatherlike solutions of the same equation with a nonzero background). We also confirm the robustness of the theoretical results by comparing the analytical predictions with careful numerical simulations with a variety of initial conditions, which confirm that the evolution of modulationally unstable media in the presence of localized initial perturbations is indeed described by the same asymptotic state.

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