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.

Evidence is presented of universal behavior in modulationally unstable media. An ensemble of nonlinear evolution equations, including three partial differential equations, an integro-differential equation, a nonlocal system and a differential-difference equation, is studied analytically and numerically. Collectively, these systems arise in a variety of applications in the physical and mathematical sciences, including water waves, optics, acoustics, Bose-Einstein condensation, and more. All these models exhibit modulational instability, namely, the property that a constant background is unstable to long-wavelength perturbations. In this work, each of these systems is studied analytically and numerically for a number of different initial perturbations of the constant background, and it is shown that, for all systems and for all initial conditions considered, the dynamics gives rise to a remarkably similar structure comprised of two outer, quiescent sectors separated by a wedge-shaped central region characterized by modulated periodic oscillations. A heuristic criterion that allows one to compute some of the properties of the central oscillation region is also given.AMS subject classifications. 35Q55, 37Kxx, 37K40, 74J30 Remarkably, the nonlinear stage of MI in the NLS equation with periodic boundary conditions is described in terms of a homoclinic structure [8,50] characterized by two qualitatively different families of recurrent (or, more strictly, doubly-periodic) solutions, through which the perturbation is cyclically amplified and back-converted to the background. These two families are separated by the so called Akhmediev breather, which represents the separatrix of the homoclinic structure, featuring a single cycle of conversion and back-conversion, with its unstable manifold corresponding to the MI linearized growth.In the more general non-periodic case (i.e., for localized perturbations of the constant background), however, the dynamics of MI is strikingly different. Indeed, using the IST for the focusing NLS equation with non-zero background [13], it was shown in [12] that in this scenario (i.e., constant boundary conditions at infinity, corresponding to localized per- †

The inverse scattering transform (IST) with non-zero boundary conditions at infinity is presented for a matrix nonlinear Schrödinger-type equation which has been proposed as a model to describe hyperfine spin F = 1 spinor Bose-Einstein condensates with either repulsive interatomic interactions and antiferromagnetic spin-exchange interactions (self-defocusing case), or attractive interatomic interactions and ferromagnetic spin-exchange interactions (self-focusing case). Both the direct and the inverse problems are formulated in terms of a suitable uniformization variable which allows to develop the IST on the standard complex plane, instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity of the scattering eigenfunctions and scattering data, symmetries, properties of the discrete spectrum, and asymptotics are derived. The inverse problem is posed as a Riemann-Hilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided. In addition, the general behavior of the one-soliton solutions is analyzed in details in the self-focusing case, including some special cases not previously discussed in the literature.

Interactions between solitons and the coherent oscillation structures generated by localized disturbances via modulational instability are studied within the framework of the focusing nonlinear Schrödinger equation. Two main interaction regimes are identified based on the relative value of the velocity of the incident soliton compared to the amplitude of the background: soliton transmission and soliton trapping. Specifically, when the incident soliton velocity exceeds a certain threshold, the soliton passes through the coherent structure and emerges on the other side with its velocity unchanged. Conversely, when the incident soliton velocity is below the threshold, once the soliton enters the coherent structure, it remains confined there forever. It is demonstrated that the soliton is not destroyed, but its velocity inside the coherent structure is different from its initial one. Moreover, it is also shown that, depending on the location of the discrete eigenvalue associated to the soliton, these phenomena can also be accompanied by the generation of additional, localized propagating waves in the coherent structure, akin to a soliton-generated wake.

We address the degree of universality of the Fermi-Pasta-Ulam recurrence induced by multisoliton fission from a harmonic excitation by analyzing the case of the semiclassical defocusing nonlinear Schrödinger equation, which models nonlinear wave propagation in a variety of physical settings. Using a suitable Wentzel-Kramers-Brillouin approach to the solution of the associated scattering problem we accurately predict, in a fully analytical way, the number and the features (amplitude and velocity) of solitonlike excitations emerging post-breaking, as a function of the dispersion smallness parameter. This also permits us to predict and analyze the near-recurrences, thereby inferring the universal character of the mechanism originally discovered for the Korteweg-deVries equation. We show, however, that important differences exist between the two models, arising from the different scaling rules obeyed by the soliton velocities.