The theory of optical dispersive shocks generated in the propagation of light beams through photorefractive media is developed. A full one-dimensional analytical theory based on the Whitham modulation approach is given for the simplest case of a sharp steplike initial discontinuity in a beam with one-dimensional striplike geometry. This approach is confirmed by numerical simulations, which are extended also to beams with cylindrical symmetry. The theory explains recent experiments where such dispersive shock waves have been observed.

Abstract. We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrödinger (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 largeamplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain spacetime 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.

The theory of nonlinear diffraction of intensive light beams propagating through photorefractive media is developed. Diffraction occurs on a reflecting wire embedded in the nonlinear medium at relatively small angle with respect to the direction of the beam propagation. It is shown that this process is analogous to the generation of waves by a flow of a superfluid past an obstacle. The "equation of state" of such a superfluid is determined by the nonlinear properties of the medium. On the basis of this hydrodynamic analogy, the notion of the "Mach number" is introduced where the transverse component of the wave vector plays the role of the fluid velocity. It is found that the Mach cone separates two regions of the diffraction pattern: inside the Mach cone oblique dark solitons are generated and outside the Mach cone the region of "ship waves" is situated. Analytical theory of "ship waves" is developed and two-dimensional dark soliton solutions of the equation describing the beam propagation are found. Stability of dark solitons with respect to their decay into vortices is studied and it is shown that they are stable for large enough values of the Mach number.

We investigate by numerical simulations the pattern formation after an oscillating attractiverepulsive obstacle inserted into the flow of a Bose-Einstein condensate. For slow oscillations we observe a complex emission of vortex dipoles. For moderate oscillations organized lined up vortex dipoles are emitted. For high frequencies no dipoles are observed but only lined up dark fragments. The results shows that the drag force turns negative for sufficiently high frequency. We also successfully model the ship waves in front of the obstacle. In the limit of very fast oscillations all the excitations of the system tend to vanish.PACS numbers: 03.75. Kk, 47.40.Ki, 05.45.Yv arXiv:1212.5644v1 [cond-mat.quant-gas]

In a previous work[1] exact stable oblique soliton solutions were revealed in two dimensional nonlinear Schrödinger flow. In this work we show that single soliton solution can be expressed within the Hirota bilinear formalism. An attempt to build two-soliton solutions shows that the system is "close" to integrability provided that the angle between the solitons is small and/or we are in the hypersonic limit. PACS numbers: 47.40.Nm, 03.75.Kk, 05.45.Yv IntroductionThe nonlinear Schrödinger (NLS) flow is ubiquitous in many physical systems such as photorefractive crystals, and the superfluids Bose-Einstein condensates and exciton-polaritons. A fundamental problem is how a superfluid reacts to the presence of an obstacle. One can define a Mach velocity (M ) as the velocity of the obstacle relative to the sound velocity in the medium. Two-dimensional studies showed that when M > 0.37 the system loses superfluidity and start to emit pair of vortices [2][3][4][5]. Increasing the velocity showed that vortices merge in a "vortex street" [6], which were later understood as oblique solitons [7] and its exact single soliton solution determined [1,8]. Oblique solitons were long know to be unstable but it was showed that under the flow they are only convectively unstable provided that M > 1.44 [9,10]. Studies with extended obstacles also presented oblique solitons in the wake, and an analytical approach based on Whitham modulation theory was successfully applied [11]. Oblique solitons were realized experimentally in the system of exciton-polaritons [12], though for lower Mach number than originally predicted by theory. Corrections to the model including losses were able to match experimental observations [13]. Dynamics of formation and decay of oblique solitons were recently observed in [14].A key question about solitons is how they behave in collisions. As long as we know, there is no general proof about the non-integrability of the 2D-NLS. Numerical studies with two obstacles were able to generate collisions between these oblique solitons [15]. These collisions were shown to be practically elastic suggesting integrability or "close" to integrability in such system. In the same work, an analytical treatment was considered using hydrodynamical approach and the system was show to follow a 1D-NLS equation in the hypersonic limit, and collisions could be described by the well known phase shifts [16]. Numerical calculations were in good agreement with predicted phase shifts, considering that they were perturbed by previous interactions with linear waves. Since the exact single soliton (1SS) was already obtained, one might ask if an exact two-soliton solution (2SS) could be found. A possible framework to find multiple soliton solutions is the Hirota method [17,18]. In the following we build up a bilinear Hirota form of the 2D-NLS in the stationary frame relative to the obstacle. Then, we show that the single oblique soliton solution indeed satisfy this form. In the sequence we propose an ansatz to the exact solution of the two-...

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