The structural features of wave collapse in medium with normal group velocity dispersion

The dynamic characteristics of self-action in three-dimensional wave packets described by the nonlinear Schrödinger equation with a hyperbolic space operator were studied analytically and numerically. The class of the initial wave field distributions for which self-focusing effects predominated over dispersion spreading and caused the arising of wave collapses was considered. The collapse of tubular wave packets was shown to be accompanied by packet shape changes during its contraction to the axis of the system. The nonlinear stabilization of collapses resulted in wave field fragmentation in the longitudinal direction followed by the expansion of the bunches thus formed along the axis. The dynamics of collapses was numerically studied taking into account medium nonlinearity saturation and nonlinear dissipation.     

Self-action dynamics of ultrashort electromagnetic pulses

The self-action dynamics of three-dimensional wave packets whose width is on the order of the carrier frequency is studied under fairly general assumptions concerning the dispersion properties of the medium. The condition for the wave field collapse is determined. Self-action regimes in a dispersion-free medium and in media with predominance of anomalous or normal group velocity dispersions are numerically investigated. It is shown that, for extremely short pulses, nonlinearity leads not only to the self-compression of the wave field but also to a “turn-over” of the longitudinal profile. In a dispersionless medium, the formation of a shock front within the pulse leads to the nonlinear dissipation of linearly polarized radiation and to self-focusing stabilization. For circularly polarized radiation, the wave collapse is accompanied by the formation of an envelope shock wave.     

Two kinds of upper hybrid soliton bistability

The basic model employed to describe nonlinear upper hybrid wave structures is the generalized nonlinear Schrödinger equation including second and fourth order dispersive effects as well as local and nonlocal nonlinearity. For two kinds of such an equation the existence of two stable solitons with the same plasmon number but with different spatial scales and amplitudes is shown as two qualitatively different kinds of upper hybrid soliton bistability. An integral relation for an arbitrary nonlinear upper hybrid wave packet evolution is derived taking into account higher order dispersive effects. Necessary conditions for soliton formation from arbitrary wave packets and the impossibility of wave packet collapse are demonstrated taking into account higher order dispersive effects.     

Regime of wave-packet self-action in a medium with normal dispersion of the group velocity

Peculiar features of the self-action of non-one-dimensional wave packets described by the nonlinear Schrodinger equation with a hyperbolic spatial operator were studied analytically and numerically. It was shown that the self-action dynamics is determined by the consequence of the processes of transverse self-focusing filamentation and longitudinal splitting. Splitting scenarios were classified. It is shown that the strongest inhomogeneities are excited along "hyperbolas" in the self-similar collapse process.     

Arresting wave collapse by wave self-rectification

We put forward a mechanism for tailoring, and even arresting, the collapse of wave packets in nonlinear media, whose dynamics is governed by nonlocal two-dimensional nonlinear Schr?dinger-like equations. The key ingredient of the scheme is the self-generation of nonlocal nonlinearities mediated by wave rectification.     

Self-Action of a Supershort Laser Pulse in a Medium with Normal Group-Velocity Dispersion

We present the results of numerical and analytical analysis of solutions of the three-dimensional (3D) nonlinear Schröodinger equation with hyperbolic spatial operator. Evolution of the system is considered in separate for two types of the initial field: a Gaussian distribution and a hollow-type (tubular or horseshoe) distribution. The effect of the nonlinear dispersion on wave-packet splitting during self-compression toward the system axis is studied. It is shown that additional focusing of Gaussian wave packets takes place in a wide range of the nonlinear-dispersion parameter. This effect results in a noticeable amplitude growth of one of the two secondary pulses formed as a result of the splitting. For hollow-type distributions, we note the formation of moving inhomogeneities and the excitation of secondary wave fields typical of the hyperbolic system.     

Electrohydrodynamic wave-packet collapse and soliton instability for dielectric fluids in (2 + 1)-dimensions

A weakly nonlinear theory of wave propagation in two superposed dielectric fluids in the presence of a horizontal electric field is investigated using the multiple scales method in (2 + 1)-dimensions. The equation governing the evolution of the amplitude of the progressive waves is obtained in the form of a two-dimensional nonlinear Schrödinger equation. We convert this equation for the evolution of wave packets in (2 + 1)-dimensions, using the function transformation method, into an exponentional and a Sinh-Gordon equation, and obtain classes of soliton solutions for both the elliptic and hyperbolic cases. The phenomenon of nonlinear focusing or collapse is also studied. We show that the collapse is direction-dependent, and is more pronounced at critical wavenumbers, and dielectric constant ratio as well as the density ratio. The applied electric field was found to enhance the collapsing for critical values of these parameters. The modulational instability for the corresponding one-dimensional nonlinear Schrödinger equation is discussed for both the travelling and standing waves cases. It is shown, for travelling waves, that the governing evolution equation admits solitary wave solutions with variable wave amplitude and speed. For the standing wave, it is found that the evolution equation for the temporal and spatial modulation of the amplitude and phase of wave propagation can be used to show that the monochromatic waves are stable, and to determine the amplitude dependence of the cutoff frequencies.Received: 23 November 2003, Published online: 15 March 2004PACS: 47.20.-k Hydrodynamic stability - 52.35.Sb Solitons; BGK modes - 42.65.Jx Beam trapping, self-focusing and defocusing; self-phase modulation - 47.65. + a Magnetohydrodynamics and electrohydrodynamicsM.F. El-Sayed: Permanent address: Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt     

激光在等离子体中的坍塌行径

从椭圆偏振激光场波包所满足的非线性控制方程出发,利用场论方法,构造该非线性方程的拉格朗日密度函数,得到激光场波包的能量集中具有准粒子特性,并给出守恒的粒子数、动量和能量。讨论了调制不稳定发展后期的波包坍塌动力学问题,利用归一化的粒子数密度分布作为权重,分析了激光场的波包。结果表明,激光场波包的尺度在有限的时间进程中,将塌缩到一个很小的值,证明了激光场具有塌缩行径。     

Dynamics of wave packets and soliton interaction in terms of the third-order nonlinear Schrödinger equation

We present the results of numerical study of the evolution of wave packets and envelope soliton interaction in terms of the third-order nonlinear Schrödinger equation. It is shown that an arbitrary initial pulse evolves to a few solitons and a linear quasiperiodic wave. The interaction of solitons is accompanied by the radiation of part of the wave field in the form of a linear quasiperiodic wave from the interaction region, amplification of the soliton with larger amplitude and attenuation of the soliton with smaller amplitude.     

Stationary wave packets in the Kerr nonlinear media with imaginary harmonic potential and linear gain

The existence of stationary wave packets in the nonlinear Kerr media with an imaginary harmonic potential and a linear gain is investigated. By employing a variational approach the existence of stable bright solitons is shown for the case of a defocusing nonlinearity. In focusing nonlinear media, the bright solitons have been shown to be unstable. The predictions of variational approach are confirmed by numerical simulations of the full modified NLS equation. The predicted stationary localized wave packets can be observed in a quasi-one-dimensional BEC with an imaginary optical potential and atoms feeding.     

Focusing of nonlinear wave groups in deep water

The freak wave phenomenon in the ocean is explained by the nonlinear dynamics of phase-modulated wave trains. It is shown that the preliminary quadratic phase modulation of wave packets leads to a significant amplification of the usual modulation (Benjamin-Feir) instability. Physically, the phase modulation of water waves may be due to a variable wind in storm areas. The well-known breather solutions of the cubic Schrödinger equation appear on the final stage of the nonlinear dynamics of wave packets when the phase modulation becomes more uniform.     

Periodic Solutions for a Class of Nonlinear Partial Differential Equations in Higher Dimension

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schrödinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases of completely resonant equations, where the bifurcation equation is infinite-dimensional, such as the nonlinear Schrödinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist.     

Nonlinear X-wave formation and conical emission at different powers of a femtosecond laser pulse in water

Nonlinear X-wave formation at different pulse powers in water is simulated using the standard model of nonlinear Schrödinger equation (NLSE). It is shown that in near field X-shape originally emerges from the interplay between radial diffraction and optical Kerr effect. At relatively low power group-velocity dispersion (GVD) arrests the collapse and leads to pulse splitting on axis. With high enough power, multi-photon ionization (MPI) and multi-photon absorption (MPA) play great importance in arresting the collapse. The tailing part of pulse is first defocused by MPI and then refocuses. Pulse splitting on axis is a manifestation of this process. Double X-wave forms when the split sub-pulses are self-focusing. In the far field, the character of the central X structure of conical emission (CE) is directly related to the single or double X-shape in the near field.     

Self-action dynamics of ultrashort electromagnetic pulses

The equation generalizing the nonlinear Schrödinger equation to the case of pulses with a duration of few field oscillation periods is analyzed. A change in the effective parameters (centroid, duration, and width) of the wave field on the pulse propagation path are determined by the moments method. The collapse of spatial structure is shown to occur, and its formation associated with the steepening of the pulse leading edge are numerically studied.     

Nonlinear dopplerons in semimetals

The possibility of the nonlinear wave propagation occurring in semimetals in the geometry where a constant magnetic field H is directed along the trigonal axis of the crystal has been investigated theoretically. In the linear regime in this geometry, there is a strong magnetic Landau damping without the wave propagation. It has been shown that the electron trapping by the magnetic field of a large-amplitude radio-frequency wave decreases the efficiency of this damping. As a result, nonlinear dopplerons can propagate in arsenic and, possibly, in antimony.     

Dynamical behaviour of transverse plasmons in pair plasmas

This paper analytically investigates the nonlinear behaviour of transverse plasmons in pair plasmas on the basis of the nonlinear governing equations obtained from Vlasov--Maxwell equations. It shows that high frequency transverse plasmons are modulationally unstable with respect to the uniform state of the pair plasma. Such an instability would cause wave field collapse into a localized region. During the collapse process, ponderomotive expulsion is greatly enhanced for the increase of wave field strength, leading to the formation of localized density cavitons which are significant for the future experimental research in the interaction between high frequency electromagnetic waves and pair plasmas.     

Electron wave packet dynamics in twisted nonlinear ladders with correlated disorder

Within a tight-binding Hamiltonian approach, we study the dynamics of one-electron wave packets in a twisted ladder geometry with adiabatic electron-phonon interaction. The electron-phonon coupling is taken into account in the time-dependent Schrödinger equation through a cubic nonlinearity. This physical scenario incorporates several relevant ingredients to study the electronic wave packet dynamics in DNA-like segments. In the absence of nonlinearity, a random sequence of nucleotides pairs makes the wave packets remain localized, according to the standard picture of the Anderson localization. However, when the electron-phonon interaction is turned on, Anderson localization is suppressed and a subdiffusive regime takes place. Further, we show that the wave packet trapping can be controlled by an external field perpendicular to the helicity axis of the double-strand chain.     

Self-focusing of wave packets and envelope shock formation in nonlinear dispersive media

The self-action of three-dimensional wave packets is analyzed analytically and numerically under the conditions of competing diffraction, cubic nonlinearity, and nonlinear dispersion (dependence of group velocity on wave amplitude). A qualitative analysis of pulse evolution is performed by the moment method to find a sufficient condition for self-focusing. Self-action effects in an electromagnetically induced transparency medium (without cubic nonlinearity) are analyzed numerically. It is shown that the self-focusing of a wave packet is accompanied by self-steepening of the longitudinal profile and envelope shock formation. The possibility of envelope shock formation is also demonstrated for self-focusing wave packets propagating in a normally dispersive medium.     

The Franz-Keldysh effect in superlattices in the field of a nonlinear electromagnetic wave

We study the effect of interminiband breakdown in a semiconductor quantum superlattice in a constant electric field and in the field of a nonlinear wave whose intensities are directed along the superlattice axis. The problem has been solved in the quasiclassical approximation for an arbitrary ratio between the widths of the allowed and forbidden minibands. In particular cases, we have obtained formulas for the breakdown probability in the presence of only one field, as well as for a linear wave and a solitary wave (soliton). It is shown that the probability of interminiband breakdown increases with the nonlinearity parameter of the electromagnetic wave k. The absorption coefficient of the nonlinear wave is calculated for typical parameters of the superlattice. Pedagogical University, Volgograd, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 6, pp. 758–766, June, 1998.     

A method for constructing radial wave packets with application to target distortion in electron-atom collisions

It has been shown that an analysis of radial stationary state wave functions of a particle in terms of their loops leads to such continuous, single-valued and finite functions which represent a practically convenient form of the radial wave packets of that particle at various positions. The radial wave packets have been used to investigate target distortion in electron-atom collisions. The distortion of the target is defined in terms of quantum-mechanical probabilities given by the wave packets. A closed expression which depends upon the position of the colliding electron, is obtained for the potential energy of the target in the field of the colliding electron.