Modulational instability in the cubic-quintic nonlinear Schrödinger equation through the variational approach

The dynamics of nonlinear pulse propagation in an average dispersion-managed soliton system is governed by a constant coefficient nonlinear Schrödinger (NLS) equation. For a special set of parameters the constant coefficient NLS equation is completely integrable. The same constant coefficient NLS equation is also applicable to optical fiber systems with phase modulation or pulse compression. We also investigate MI arising in the cubic-quintic nonlinear Schrödinger equation for ultrashort pulse propagation. Within this framework, we derive ordinary differential equations (ODE’s) for the time evolution of the amplitude and phase of modulation perturbations. Analyzing the ensuing ODE’s, we derive the classical modulational instability criterion and identify it numerically. We show that the quintic nonlinearity can be essential for the stability of solutions. The evolutions of modulational instability are numerically investigated and the effects of the quintic nonlinearity on the evolutions are examined. Numerical simulations demonstrate the validity of the analytical predictions.     

Spectral stability analysis for periodic traveling wave solutions of NLS and CGL perturbations

We consider cnoidal traveling wave solutions to the focusing nonlinear Schrödinger equation (NLS) that have been shown to persist when the NLS is perturbed to the complex Ginzburg-Landau equation (CGL). We show that while these periodic traveling waves are spectrally stable solutions of NLS with respect to periodic perturbations, they are unstable with respect to bounded perturbations. Furthermore, we use an argument based on the Fredholm alternative to find an instability criterion for the persisting solutions to CGL.     

Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrödinger equation

The modulational instability of perturbed plane-wave solutions of the cubic nonlinear Schrödinger (NLS) equation is examined in the presence of three forms of dissipation. We present two families of decreasing-in-magnitude plane-wave solutions to this dissipative NLS equation. We establish that all such solutions that have no spatial dependence are linearly stable, though some perturbations may grow a finite amount. Further, we establish that all such solutions that have spatial dependence are linearly unstable if a certain form of dissipation is present.     

Nonlinear fast growth of water waves under wind forcing

In the wind-driven wave regime, the Miles mechanism gives an estimate of the growth rate of the waves under the effect of wind. We consider the case where this growth rate, normalised with respect to the frequency of the carrier wave, is of the order of the wave steepness. Using the method of multiple scales, we calculate the terms which appear in the nonlinear Schrödinger (NLS) equation in this regime of fast-growing waves. We define a coordinate transformation which maps the forced NLS equation into the standard NLS with constant coefficients, that has a number of known analytical soliton solutions. Among these solutions, the Peregrine and the Akhmediev solitons show an enhancement of both their lifetime and maximum amplitude which is in qualitative agreement with the results of tank experiments and numerical simulations of dispersive focusing under the action of wind.     

Numerical modeling of spatio-temporal solitons using an adaptive spherical Fourier Bessel split-step method

We present a novel technique to numerically solve pulsed optical beam or “light bullet” propagation in bulk nonlinear media based on the scalar nonlinear Schrödinger (NLS) equation in three dimensions with spherical symmetry. Using fast algorithms for spherical Fourier Bessel transforms along with adaptive longitudinal stepping and transverse grid management in a symmetrized split-step technique, it is possible to accurately study many nonlinear effects, including the possibility of spatio-temporal collapse, or the collapse-arresting mechanism due to saturable nonlinearity.     

On the description of quadratic and cubic optical effects of propagation of femtosecond pulses within the framework of the model of the saturable potential

It is shown that, for describing the simultaneous manifestation of nonlinear effects of different orders (quadratic, cubic, etc.) in propagation of high-intensity femtosecond pulses, it is sufficient to use the saturable potential of the restoring force. With the use of computer simulation, it is demonstrated that the spectrum of the response of a medium calculated on the basis of the Duffing equation with quadratic and cubic nonlinearity is in qualitative agreement with the spectrum calculated on the basis of the equation with a saturable restoring force. The dependence of the frequency corresponding to the maximum spectral radiance on the duration of the interacting pulse is discussed. It is noted that, for ultrashort pulses, even in the case of their propagation in a linear medium, the response can occur at the natural frequency rather than at the frequency of the propagating pulse.     

Evolution of optical pulses in the presence of third-order dispersion

We model the propagation of femtosecond pulses through optical fibres by a nonlinear Schrödinger (NLS) equation involving a perturbing term arising due to third-order dispersion in the medium. The perturbative effect of this higher-order dispersion causes the usual NLS soliton to emit a radiation field. As a result, the given initial pulse propagating through the fibre exhibits nonsolitonic behaviour. We make use of a variational method to demonstrate how an initial pulse by the interaction with the emitted radiation can evolve into a soliton. We also demonstrate that the effect of interaction between the initial pulse and radiation field can be accounted for by including, in the evolution equation, terms associated with self-steepening and stimulated Raman scattering that characterize the optical medium.     

Propagation of a cos-Gaussian beam in a Kerr medium

The dynamic properties of cos-Gaussian beams in the presence of Kerr nonlinearity are investigated analytically and numerically using the nonlinear Schrödinger equation (NLS). Based on the moments method, evolution of a cos-Gaussian beam width in the root-mean-square (RMS) sense is obtained analytically. The beam propagation factors and the critical powers of the cos-Gaussian beams with a uniform wavefront are calculated. Using numerical simulation, it is found that although the RMS beam width broadens, the central parts of the beam give rise to an initial radial compression and a significant redistribution during propagation. The partial collapse of central parts of the beam is observed below the threshold for a global collapse. The cos-Gaussian beams eventually convert into cosh-Gaussian type beams in Kerr media with low initial power.     

Solitonen in Wellenleitern mit nichtlinearer optischer Beschichtung

Solitons in a Waveguide with a Thin Nonlinear Optical Slab Starting from Maxwell's equations the influence of a thin nonlinear optical slab embed in a waveguide on the propagation of a monomode (TE) pulse is investigated. The amplitude of the pulse is described by a nonlinear Schrödinger equation (NLS) containing parameters of the waveguide with the nonlinear slab and leading to solitons in the nonlinear modified waveguide.     

强双折射光纤中单一频率传输区域的调制不稳定性

利用光脉冲在光纤中传播时所遵守的相干非线性薛定谔耦合方程,研究了单一频率的线偏振光,且偏振方向沿两个双折射轴的分量强度相等时,在反常色散区和正常色散区所产生的调制不稳定性.结果表明在反常色散区和正常色散区,对应不同的功率区域输入脉冲有不同的增益谱,并且当输入脉冲功率一定时,双折射性质变化导致增益谱表现出明显的不同.     

超高斯型光脉冲在色散管理光纤系统中的传输

从描述超短光脉冲在光纤中传输的高阶非线性薛定谔方程入手,使用傅里叶变换的分裂算符法,通过计算机模拟,研究了超高斯光脉冲在色散管理光纤系统中的传输特性,并考察了超高斯光脉冲在加入随机噪声后传输的稳定性以及脉冲间的相互作用.     

Dark Solitons for the Defocusing Cubic Nonlinear Schrödinger Equation with the Spatially Periodic Potential and Nonlinearity

We study the existence of dark solitons of the defocusing cubic nonlinear Schrödinger (NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincaré map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity.     

Theoretical and experimental study of blurring of a femtosecond laser pulse in a turbid medium

We propose an analytical model of the spatio-temporal structure of a short laser pulse transmitted through a layer of an optically inhomogeneous medium with high anisotropy of scattering. The light-field brightness in the medium is represented as a finite series in terms of multiplicities of the small-angle scattering, while the contribution from the higher-order scattering is allowed for as a quasi-diffuse component. The scattered-pulse structure is calculated on the basis of solving the radiative-transfer equation in the small-angle approximation with allowance for the effect of multipath light propagation. Compared with the first approximation of the multiple-scattering theory (attenuated nonscattered light plus the diffuse component), this approach makes it possible to describe more correctly the transformation of the spatio-angular distribution of light in the medium when passing from the single-scattering to multiple-scattering regime, as well as specify the temporal profile of the scattered pulse. The temporal profile of the femtosecond pulse transmitted through a layer of model scattering medium with various concentrations of scatterers is studied experimentally. The blurred-pulse structure is studied with the help of nonlinear optical gating in the case of noncollinear generation of the second harmonic. Good agreement between the theoretical and experimental time profiles of the scattered pulse is shown for the optical-thickness intervals corresponding to both the predominantly low multiplicity scattering and multiple small-angle scattering, which allows us to use the proposed analytical model for solving the inverse problem of the pulse sounding of a homogeneous turbid medium. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 4, pp. 333–348, April 2008.     

双折射光纤中斯托克斯波的增益谱

根据激光脉冲在双折射光纤中传输时的混合模方程,考虑拉曼效应后,得出在受激拉曼散射和参量放大共同作用下的增益。数值模拟了当输入泵浦波沿两双折射轴450附近区域传输时,在不同色散区斯托克斯增益谱随输入功率及群速度失配等相关参数的变化关系;结果表明斯托克斯增益谱在不同色散区表现出明显不同的特性,当满足一定条件下,可利用混合模双折射光纤中传输的泵浦波分解出的脉冲序列来分离和提取T频脉冲。     

双折射光纤中斯托克斯波的增益谱

根据激光脉冲在双折射光纤中传输时的混合模方程,考虑拉曼效应后,得出在受激拉曼散射和参量放大共同作用下的增益。数值模拟了当输入泵浦波沿两双折射轴450附近区域传输时,在不同色散区斯托克斯增益谱随输入功率及群速度失配等相关参数的变化关系;结果表明斯托克斯增益谱在不同色散区表现出明显不同的特性,当满足一定条件下,可利用混合模双折射光纤中传输的泵浦波分解出的脉冲序列来分离和提取T频脉冲。     

One-parameter localized traveling waves in nonlinear Schrödinger lattices

We address the existence of traveling single-humped localized solutions in the spatially discrete nonlinear Schrödinger (NLS) equation. A mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero. The technique is based on using the Implicit Function Theorem for solution of an appropriate differential advance-delay equation in exponentially weighted spaces. The resulting Melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse, which are checked numerically. We apply the technique to the so-called Salerno model and the translationally invariant discrete NLS equation with a cubic nonlinearity. We show that the traveling solutions terminate in the Salerno model whereas they generally persist in the translationally invariant NLS lattice as a one-parameter family of solutions. These results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails. Analysis of persistence also predicts the spectral stability of the one-parameter family of traveling solutions under time evolution of the discrete NLS equation.     

Nonlinearity management in optics: experiment, theory, and simulation

We conduct an experimental investigation of nonlinearity management in optics using femtosecond pulses and layered Kerr media consisting of glass and air. By examining the propagation properties over several diffraction lengths, we show that wave collapse can be prevented. We corroborate these experimental results with numerical simulations of the (2+1)-dimensional focusing cubic nonlinear Schr?dinger equation with piecewise constant coefficients and a theoretical analysis of this setting using a moment method.     

Role of dispersion in multiple-collapse dynamics

The multiple-collapse dynamics of ultrashort pulses along the propagation direction are investigated under conditions of both normal and anomalous group-velocity dispersion (GVD). In the anomalous-GVD regime we find that collapse events can occur at locations in the medium many diffraction lengths beyond the initial collapse point, in contrast with the normal-GVD regime in which multiple collapse occurs within a diffraction length. Numerical simulations of a modified nonlinear envelope equation are found to be in good qualitative agreement with the observed lengths of the filaments.     

非线性双折射色散光纤中极化调制不稳定性分析

利用光脉冲在光纤中传播时所遵守的相干非线性薛定谔耦合方程,研究了线偏振光在光纤中的传输特性.结果表明:由于线性双折射、非线性效应和色散的相互作用,导致光的偏振状态发生变化,产生交叉相位调制(XPM),从而导致调制不稳定性.这一调制不稳性不仅在反常色散区产生,在正常色散区也能产生,并且线性双折射发生变化时,增益谱也随之发生变化.