The Integrable Nature of Modulational Instability

We investigate the nonlinear stage of the modulational (or Benjamin–Feir) instability by characterizing the initial value problem for the focusing nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBC) at infinity. We do so using the recently formulated inverse scattering transform (IST) for this problem. While the linearization of the NLS equation ceases to be valid when the perturbations have grown sufficiently large compared to the background, the results of the IST remain valid for all times and therefore provide a convenient way to study the nonlinear stage of the modulational instability. We begin by studying the spectral problem for the Dirac operator (i.e., the first half of the Lax pair for the NLS equation) with piecewise constant initial conditions which are a generalization to NZBC of a potential well and a potential barrier. Since the scattering data uniquely determine the time evolution of the initial condition via the inverse problem, the study of these kinds of potentials provides a simple means of investigating the growth of small perturbations of a constant background via IST. We obtain several results. First, we prove that there are arbitrarily small perturbations of the constant background for which there are discrete eigenvalues, which shows that no area theorem is possible for the NLS equation with NZBC. Second, we prove that there is a class of perturbations for which no discrete eigenvalues are present. In particular, this latter result shows that solitons cannot be the primary vehicle for the manifestation of the instability, contrary to a recent conjecture. We supplement these results with a numerical study about the existence, number, and location of discrete eigenvalues in other situations. Finally, we compute the small-deviation limit of the IST, and we compare it with the direct linearization of the NLS equation around a constant background, which allows us to precisely identify the nonlinear analogue of the unstable Fourier modes within the IST. These are the Jost eigenfunctions for values of the scattering parameter belonging to a finite interval of the imaginary axis around the origin. Importantly, this last result shows that the IST contains an automatic mechanism for the saturation of the modulational instability.