Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation

Solitons and breathers are localized solutions of integrable systems that can be viewed as “particles” of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media, these “integrable” gases present a fundamental interest for nonlinear physics. We develop an analytical theory of breather and soliton gases by considering a special, thermodynamic-type limit of the wave-number–frequency relations for multiphase (finite-gap) solutions of the focusing nonlinear Schrödinger equation. This limit is defined by the locus and the critical scaling of the band spectrum of the associated Zakharov-Shabat operator, and it yields the nonlinear dispersion relations for a spatially homogeneous breather or soliton gas, depending on the presence or absence of the “background” Stokes mode. The key quantity of interest is the density of states defining, in principle, all spectral and statistical properties of a soliton (breather) gas. The balance of terms in the nonlinear dispersion relations determines the nature of the gas: from an ideal gas of well separated, noninteracting breathers (solitons) to a special limiting state, which we term a breather (soliton) condensate, and whose properties are entirely determined by the pairwise interactions between breathers (solitons). For a nonhomogeneous breather gas, we derive a full set of kinetic equations describing the slow evolution of the density of states and of its carrier wave counterpart. The kinetic equation for soliton gas is recovered by collapsing the Stokes spectral band. A number of concrete examples of breather and soliton gases are considered, demonstrating the efficacy of the developed general theory with broad implications for nonlinear optics, superfluids, and oceanography. In particular, our work provides the theoretical underpinning for the recently observed remarkable connection of the soliton gas dynamics with the long-term evolution of spontaneous modulational instability

Semiclassical dynamics of quasi-one-dimensional, attractive Bose-Einstein condensates

The strongly interacting regime for attractive Bose-Einstein condensates (BECs) tightly confined in an extended cylindrical trap is studied. For appropriately prepared, non-collapsing BECs, the ensuing dynamics are found to be governed by the one-dimensional focusing Nonlinear Schr\"odinger equation (NLS) in the semiclassical (small dispersion) regime. In spite of the modulational instability of this regime, some mathematically rigorous results on the strong asymptotics of the semiclassical limiting solutions were obtained recently. Using these results, "implosion-like" and "explosion-like" events are predicted whereby an initial hump focuses into a sharp spike which then expands into rapid oscillations. Seemingly related behavior has been observed in three-dimensional experiments and models, where a BEC with a sufficient number of atoms undergoes collapse. The dynamical regimes studied here, however, are not predicted to undergo collapse. Instead, distinct, ordered structures, appearing after the "implosion", yield interesting new observables that may be experimentally accessible.Comment: 9 pages, 3 figure

Breaking of Symmetrical Periodic Solutions In A Singularly Perturbed Kdv Model

There are several recent developments in the well-known problem of breaking of homoclinic orbits (splitting of separatrices) of a system that undergoes a singular perturbation. First, survival of a homoclinic orbit is an exceptional situation that can be linked to triviality of the Stokes phenomenon of the underlying truncated equation. Second, homoclinic connections to exponentially small periodic orbits survive the perturbation in the generic case. In this paper we consider a different problem: we study deformations of genuine periodic orbits of the second order equation y \u27\u27 = y + y(2) that undergoes the singular perturbation epsilon(2)y \u27\u27\u27\u27 + (1 - epsilon(2))y \u27\u27 = y + y(2), where epsilon \u3e 0 is a small parameter. We prove that if the period and the constant of motion do not change too rapidly (in epsilon), a genuine (nontrivial) periodic solution does not survive the perturbation

Dam break problem for the focusing nonlinear Schr\"odinger equation and the generation of rogue waves

We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schr\"odinger (NLS) equation with the initial condition in the form of a rectangular barrier (a "box"). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains --- the dispersive dam break flows --- generated in the NLS box problem. We show that the interaction dynamics results in the emergence of modulated large-amplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain space-time domain. Our semi-classical analytical results are shown to be in excellent agreement with the results of direct numerical simulations of the small-dispersion focusing NLS equation.Comment: 29 pages, 15 figures, major revisio