We investigate propagating dark soliton solutions of the two-dimensional defocusing nonlinear Schrodinger or Gross-Pitaevskii (NLS-GP) equation that are transversely confined to propagate in an infinitely long channel. Families of single, vortex, and multilobed solitons are computed using a spectrally accurate numerical scheme. The multilobed solitons are unstable to small transverse perturbations. However, the single-lobed solitons are stable if they are sufficiently confined along the transverse direction, which explains their effective one-dimensional dynamics. The emergence of a transverse modulational instability is characterized in terms of a spectral bifurcation. The critical confinement width for this bifurcation is found to coincide with the existence of a propagating vortex solution and the onset of a "snaking" instability in the dark soliton dynamics that, in turn, give rise to vortex or multivortex excitations. These results shed light on the superfluidic hydrodynamics of dispersive shock waves in Bose-Einstein condensates and nonlinear optics.

We suggest a method for measuring the polarization orientation of high-intensity beams, by analyzing the damage structures on metal targets, created by laser-plasma-metal interaction. We apply our method on laser filaments and demonstrate stability and instability of the polarization orientation dynamics. Our experimental results show that the polarization orientation of linearly polarized input beams during filamentation is stable, whereas that of elliptically polarized input beams is not. The results are supported by an analytical model.

The nature of transverse instabilities of dark solitons for the (2+1)-dimensional defocusing nonlinear Schrodinger/Gross-Pitaevskii (NLS/GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS/GP equation. The dispersion relation for shallow solitons is obtained asymptotically beyond the KP limit. This yields (1) the maximal growth rate and associated wavenumber of unstable perturbations and (2) the separatrix between convective and absolute instabilities. The instability properties of the dark soliton are directly related to those of oblique dispersive shock wave (DSW) solutions. Stationary and nonstationary oblique DSWs are constructed analytically and investigated numerically by direct simulations of the NLS/GP equation. It is found that stationary and nonstationary oblique DSWs have the same jump conditions in the shallow and hypersonic regimes. These results have application to controlling nonlinear waves in dispersive media.

We consider a class of nonlinear Schrodinger / Gross-Pitaevskii (NLS/GP) equations with periodic potentials, having an even symmetry. We construct "solitons", centered about any point of symmetry of the potential. For focusing (attractive) nonlinearities, these solutions bifurcate from the zero state at the lowest band edge frequency, into the semi-infinite spectral gap. Our results extend to bifurcations into finite spectral gaps, for focusing or defocusing (repulsive) nonlinearities under more restrictive hypotheses. Soliton nonlinear bound states with frequencies near a band edge are well-approximated by a slowly decaying solution of a homogenized NLS/GP equation, with constant homogenized effective mass tensor and effective nonlinear coupling coefficient, modulated by a Bloch state. For the critical NLS equation with a periodic potential, e.g. the cubic two dimensional NLS/GP with a periodic potential, our results imply that the limiting soliton power, as the spectral band edge frequency is approached, is equal to a constant \zeta_* times the minimal mass soliton of the translation invariant critical NLS equation. \zeta_* is expressible in terms of the band edge Bloch eigenfunction and the determinant of the effective mass tensor; and 0<\zeta_*<1$ for any non-constant potential. The results are confirmed by numerical computation of bound states with frequencies near the spectral band edge. Finally, these results have implications for the control of nonlinear waves using periodic structures.

Dispersive shock waves (DSWs) are studied theoretically in the context of two-dimensional (2D) supersonic flow of a superfluid. Employing Whitham averaging theory for the repulsive Gross-Pitaevskii (GP) equation, suitable jump and entropy conditions are obtained for an oblique DSW, a fundamental building block for 2D flows with boundaries. In analogy to oblique viscous shock waves (VSWs), these conditions yield analytic relations between Mach number (M), velocity deflection angle (theta), and wave angle (beta). Unlike VSWs, the M-theta-beta phase diagram for DSWs displays four distinct regions associated with phase transitions in supersonic flow over a corner which are predicted and verified by numerical computations of the GP equation. Quasistationary DSWs, shock detachment due to transonic flow, spontaneous excitation of vortices, and the onset of turbulent dynamics associated with cavitation of the superfluid are observed.

We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multidimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasiperiodic, single waveguide, etc. We show that when the soliton is unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to a focusing instability, whereas violation of the spectral condition leads to a drift instability. We also present a quantitative approach that allows one to predict the stability and instability strength.

We derive an analytic formula for the lateral dynamics of solitons in a general inhomogeneous nonlinear media and demonstrate numerically that it can be valid for tens of diffraction lengths.

We derive an analytic formula for the lateral dynamics of solitons in a general inhomogeneous nonlinear media, and show that it can be valid over tens of diffraction lengths. In particular, we show that solitons centered at a lattice maximum can be "mathematically unstable" but "physically stable." We also derive an analytic upper bound for the critical velocity for tunneling, which is valid even when the standard Peierls-Nabarro potential approach fails.

We show that violation of the "spectral condition" leads to a drift in stability of a soliton from its initial location, and show how to determine the speed of the drift.

A similar type of nonlocal nonlinear Schrödinger (NLS)system arises in both water waves and nonlinear optics. Thenonlocality is due to a coupling between the first harmonic and amean term. These systems are termed nonlinear Schrödinger withmean or NLSM systems. They were first derived in water waves byBenney-Roskes and later by Davey-Stewartson. Subsequently similarequations were derived and found to be fundamental systems inquadratically nonlinear optical media. Wave collapse can occur inthese systems. The collapse structure and the role of the groundstate in the collapse process are studied. There are similarities tothe well-known collapse mechanism associated with classical NLSsystem. Numerical simulations show that NLSM collapse occurs with aquasi self-similar profile that is a modulation of the correspondingground-state. Further, it is found that NLSM collapse can bearrested by adding small nonlinearsaturation.

We compute and study localized nonlinear modes (solitons) in the semi-infinite gap of the focusing two-dimensional nonlinear Schrödinger (NLS) equation with various irregular lattice-type potentials. The potentials are characterized by large variations from periodicity, such as vacancy defects, edge dislocations, and a quasicrystal structure. We use a spectral fixed-point computational scheme to obtain the solitons. The eigenvalue dependence of the soliton power indicates parameter regions of self-focusing instability; we compare these results with direct numerical simulations of the NLS equation. We show that in the general case, solitons on local lattice maximums collapse. Furthermore, we show that the Nth-order quasicrystal solitons approach Bessel solitons in the large-N limit.

ДВУМЕРНЫЕ СОЛИТОНЫ В НЕРЕГУЛЯРНЫХ РЕШЕТОЧНЫХ СИСТЕМАХЛокализованные нелинейные моды (солитоны) рассчитаны и исследованы в случае полубесконечной щели спектра фокусирующего двумерного нелинейного уравнения Шредингера с различными нерегулярными потенциалами решеточного типа.Потенциалы характеризуются такими существенными отклонениями от периодичности, как дефекты вакансий, краевые дислокации, а также квазикристаллическая структура.Расчет солитонов ведется на основании спектральной вычислительной схемы с неподвижной точкой.Зависимость мощности солитона от собственных

A nonlinear model of spin-wave excitation using a point contact in a thin ferromagnetic film is introduced. Large-amplitude magnetic solitary waves are computed, which help explain recent spin-torque experiments. Numerical simulations of the fully nonlinear model predict excitation frequencies in excess of 0.2 THz for contact diameters smaller than 6 nm. Simulations also predict a saturation and redshift of the frequency at currents large enough to invert the magnetization under the point contact. The theory is approximated by a cubic complex Ginzburg-Landau type equation. The mode's nonlinear frequency shift is found by use of perturbation techniques, whose results agree with those of direct numerical simulations.

The relation between the fundamental parameters of energy and temporal duration of ultrashort pulses, under the condition of varying the average dispersion, are demonstrated both theoretically and experimentally in a solid-state femtosecond mode-locked laser. An asymptotic theory for nonlinear and dispersion managed solitons agrees well with the experimental data and demonstrates that the dominant factor in the pulse dynamics arises from the equilibrium established between the nonlinear Kerr effect and linear dispersion.

We show experimentally that the spatial profile of a collapsing beam evolves to the circularly symmetric Townes profile. We also show deterministic methods for controlling multiple filamentation of high-power pulses. The basic model for propagation of intense laser beams in a bulk Kerr medium is given by the dimensionless 2D nonlinear Schrödinger equation (NLS) 2 ( , , ) 0. z xx yy i x y z A A A A A + + + = Here A is the amplitude of the electric field, z is the propagation direction, and x and y are the transverse directions. The NLS has waveguide solutions of the A(z,r) = a exp(i a z) R(ar), where r = (x+y), and R, the so-called Townes profile, is the ground-state solution of the equation

We show numerically for continuous-wave beams and experimentally for femtosecond pulses propagating in air, that the collapse distance of intense laser beams in a bulk Kerr medium scales as 1/P;1/2 for input powers P that are moderately above the critical power for self focusing, but that at higher powers the collapse distance scales as 1/P.

We show that small negative fourth-order dispersion can arrest spatiotemporal collapse of ultrashort pulses with anomalous dispersion in a planar waveguide with pure Kerr nonlinearity, resulting in (2 + 1)D optical bullets. Similarly to solitons, these bullets undergo elastic collisions. Since these bullets can self-trap from noisy Gaussian input beams and propagate without any power losses, this result may be used to realize experimentally stable, nondissipative optical bullets.

In this Letter we provide what is believed to be the first experimental evidence of suppression of the number of filaments for high-intensity laser pulses propagating in air by beam astigmatism. We also show that the number, pattern, and spatial stability of the filaments can be controlled by varying the angle that a focusing lens makes with the axial direction of propagation. This new methodology can be useful for applications involving atmospheric propagation, such as remote sensing.

The carrier-envelope phase slip of an ultrashort pulse circulating in a mode-locked Ti:sapphire laser is analyzed. The laser cavity is modeled by a dispersion- and nonlinearity-managed nonlinear Schrödinger equation. The combined contributions to the phase slip induced by nonlinear phase and nonlinear dispersion are found to approach zero for strong dispersion maps. The dependence of the slip on third-order dispersion is found as well. The analytical results are verified using numerical simulations.

We derive a perturbed two-dimensional nonlinear Schrodinger equation which describes the propagation of gap-soliton bullets in nonlinear periodic waveguides at frequencies close to the gap for Bragg reflection. Analysis and simulations of this equation show that the bullets amplitude undergoes stable fOCLIsing-defocusing cycles. (C) 2003 Elsevier B.V. All rights reserved.