Statistics of Extreme Events in Integrable Turbulence.

We use the spectral kinetic theory of soliton gas to investigate the likelihood of extreme events in integrable turbulence described by the one-dimensional focusing nonlinear Schrödinger equation (fNLSE). This is done by invoking a stochastic interpretation of the inverse scattering transform for fNLSE and analytically evaluating the kurtosis of the emerging random nonlinear wave field in terms of the spectral density of states of the corresponding soliton gas. We then apply the general result to two fundamental scenarios of the generation of integrable turbulence: (i) the asymptotic development of the spontaneous modulational instability of a plane wave, and (ii) the long-time evolution of strongly nonlinear, partially coherent waves.

Topological Constraints on the Dynamics of Vortex Formation in a Two-Dimensional Quantum Fluid.

We present experimental and theoretical results on formation of quantum vortices in a laser beam propagating in a nonlinear medium. Topological constrains richer than the mere conservation of vorticity impose an elaborate dynamical behavior to the formation and annihilation of vortex-antivortex pairs. We identify two such mechanisms, both described by the same fold-Hopf bifurcation.

Dispersive Hydrodynamics of Soliton Condensates for the Korteweg-de Vries Equation.

We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special "condensate" limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the -phase KdV-Whitham modulation equations derived by Flaschka et al. (Commun Pure Appl Math 33(6):739-784, 1980) and Lax and Levermore (Commun Pure Appl Math 36(5):571-593, 1983).

Soliton gas in bidirectional dispersive hydrodynamics.

The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the soliton gas properties are determined by the overtaking elastic pairwise interactions between solitons. In this paper, we extend this theory to soliton gases in bidirectional integrable Eulerian systems where both head-on and overtaking collisions of solitons take place. We distinguish between two qualitatively different types of bidirectional soliton gases: isotropic gases, in which the position shifts accompanying the head-on and overtaking soliton collisions have the same sign, and anisotropic gases, in which the position shifts for head-on and overtaking collisions have opposite signs.

Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate.

We provide a classification of the possible flows of two-component Bose-Einstein condensates evolving from initially discontinuous profiles. We consider the situation where the dynamics can be reduced to the consideration of a single polarization mode (also denoted as "magnetic excitation") obeying a system of equations equivalent to the Landau-Lifshitz equation for an easy-plane ferromagnet. We present the full set of one-phase periodic solutions.