Dispersion management for solitons in a Korteweg-de Vries system

The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.     

矩形波导中两层流体界面上的非传播孤立波

本文用多重尺度微扰方法研究了矩形波导中二水平固壁间两层不可压无粘流体界面上重力-表面张力波的非线性调制。将Larraza和Putterman的理论进行了推广,并计入了表面张力,得出了第一阶调制波满足非线性Schrodinger方程,求出了非传播孤波解,并对结果进行了讨论。 关键词：     

Modulational Instability of Dust Ion Acoustic Waves in a Collisional Dusty Plasma

The modulational instability of dust ion acoustic waves in a dust plasma with ion-dust collision effects is studied. Using the perturbation method, a modified nonlinear Schrodinger equation contains a damping term that comes from the effect of the ion-dust collision is derived. It is found that the inclusion of the ion-dust collision would modify the modulational instability of the wave packet and could not admit any stationary envelope solitary waves.     

New types of solitary wave solutions for the higher order nonlinear Schrodinger equation

We present new types of solitary wave solutions for the higher order nonlinear Schrodinger (HNLS) equation describing propagation of femtosecond light pulses in an optical fiber under certain parametric conditions. Unlike the reported solitary wave solutions of the HNLS equation, the novel ones can describe bright and dark solitary wave properties in the same expressions and their amplitude may approach nonzero when the time variable approaches infinity. In addition, such solutions cannot exist in the nonlinear Schrodinger equation. Furthermore, we investigate the stability of these solitary waves under some initial pertubations by employing the numerical simulation methods.     

Structures of Equatorial Envelope Rossby Wave Under a Generalized External Forcing

The cubic nonlinear Schrodinger (NLS for short) equation with a generalized external heating source is derived for large amplitude equatorial envelope Rossby wave in a shear flow. And then various periodic structures for these equatorial envelope Rossby waves are obtained with the help of a new transformation, Jacobi elliptic functions,and elliptic equation. It is shown that different types of resonant phase-locked diabatic heating play different roles in structures of equatorial envelope Rossby wave.     

Periodic Structure of Equatorial Envelope Rossby Wave Under Influence of Diabatic Heating

A simple shallow-water model with influence of diabatic heating on a β-plane is applied to investigate the nonlinear equatorial Rossby waves in a shear flow. By the asymptotic method of multiple scales, the cubic nonlinear Schro^edinger (NLS for short) equation with an external heating source is derived for large amplitude equatorial envelope Rossby wave in a shear flow. And then various periodic structures for these equatorial envelope Rossby waves are obtained with the help of Jacob/elliptic functions and elliptic equation. It is shown that phase-locked diabatic heating plays an important role in periodic structures of rational form.     

Nonlinear interference between two sets of frequency conversion processes with widely spaced pump wavelengths in an optical fiber

The processes of nonlinear interaction of two high-power pump waves with widely spaced frequencies are studied in an optical fiber. Substantially increased output power of the Stokes components, generated via stimulated Raman scattering or stimulated four-wave mixing by one of the pump waves have been obtained in the field of the other pump wave and its Stokes components. For the first time the possibility to excited new spectral lines in the spectrum of one of the pump waves with frequency shifts characteristic for the other one, is experimentally demonstrated.     

Dynamics of the plane and solitary waves in a Noguchi network:Effects of the nonlinear quadratic dispersion

We consider a modified Noguchi network and study the impact of the nonlinear quadratic dispersion on the dynamics of modulated waves. In the semi-discrete limit, we show that the dynamics of these waves are governed by a nonlinear cubic Schrodinger equation. From the graphical analysis of the coefficients of this equation, it appears that the nonlinear quadratic dispersion counterbalances the effects of the linear dispersion in the frequency domain. Moreover, we establish that this nonlinear quadratic dispersion provokes the disappearance of some regions of modulational instability in the dispersion curve compared to the results earlier obtained by Pelap et al.(Phys. Rev. E 91 022925(2015)). We also find that the nonlinear quadratic dispersion limit considerably affects the nature, stability, and characteristics of the waves which propagate through the system. Furthermore, the results of the numerical simulations performed on the exact equations describing the network are found to be in good agreement with the analytical predictions.     

Rogue Waves in the (2+1)-Dimensional Nonlinear Schrodinger Equation with a Parity-Time-Symmetric Potential

The(2+1)-dimension nonlocal nonlinear Schrodinger(NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equation introduced by Ablowitz et al. [Phys. Rev. Lett. 110(2013) 064105]. General periodic solutions are derived by the bilinear method. These periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again. By taking long wave limits of the obtained periodic solutions, rogue waves are obtained. It is also shown that these line rogue waves arise from the constant background with a line profile and disappear into the constant background again in the(x,y) plane.     

Optical Solitary Waves in Fourth-Order Dispersive Nonlinear Schr(o)dinger Equation with Self-steepening and Self-frequency Shift

By making use of the generalized sine-Gordon equation expansion method, we find cnoidal periodic wave solutions and fundamental bright and dark optical solitarywave solutions for the fourth-order dispersive and the quintic nonlinear Schrodinger equation with self-steepening, and self-frequency shift. Moreover, we discuss the formation conditions of the bright and dark solitary waves.     

Evolution of basic equations for nearshore wave field

In this paper, a systematic, overall view of theories for periodic waves of permanent form, such as Stokes and cnoidal waves, is described first with their validity ranges. To deal with random waves, a method for estimating directional spectra is given. Then, various wave equations are introduced according to the assumptions included in their derivations. The mild-slope equation is derived for combined refraction and diffraction of linear periodic waves. Various parabolic approximations and time-dependent forms are proposed to include randomness and nonlinearity of waves as well as to simplify numerical calculation. Boussinesq equations are the equations developed for calculating nonlinear wave transformations in shallow water. Nonlinear mild-slope equations are derived as a set of wave equations to predict transformation of nonlinear random waves in the nearshore region. Finally, wave equations are classified systematically for a clear theoretical understanding and appropriate selection for specific applications.     

Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

Compact envelope dark solitary wave in a discrete nonlinear electrical transmission line

We introduce an extended nonlinear Schrödinger (ENLS) equation describing the dynamics of modulated waves in a nonlinear discrete electrical transmission line (NLTL) with nonlinear dispersion. We show that this equation admits envelope dark solitary wave with compact support, with width and speed independent of the amplitude, as a solution. Analytical criteria of existence and stability of this solution are derived. In particular, we show that the modulated compact wave may exist in the NLTL depending on the frequency range of the chosen carrier wave, for physically realistic parameters. The stability of compact dark solitary wave is confirmed by numerical simulations of this ENLS equation and the exact equations of the network.     

暗怪波的相互作用

以耦合非线性薛定谔方程为理论模型,数值研究了两个一阶暗怪波在正常色散单模光纤中的相互作用.基于一阶暗怪波精确解,采用分步傅里叶数值模拟法,从间距、相位差和振幅系数比方面讨论相邻两个一阶暗怪波之间的相互作用.基于二阶暗怪波精确解,讨论了两个一阶暗怪波的非线性相互作用.研究结果表明:同相位情况下,间距参数T1为0、5、20时,相邻两个一阶暗怪波相互作用激发产生“扭结型”暗怪波.相比较于单个暗怪波发生能量的弥散,“扭结型”暗怪波分裂形成多个次暗怪波.反相位情况下,间距参数T1为2、7、12时,相邻两个一阶暗怪波相互作用也可以激发产生“扭结型”暗怪波.并且“扭结型”暗怪波初始激发的空间位置偏离原始单个暗怪波的位置5.振幅系数比越大,该空间位置越接近5.二阶暗怪波可以看作是两个一阶暗怪波的非线性叠加,复合型和三组分型二阶暗怪波与相邻两个一阶暗怪波的相互作用略有相似.     

Rogue waves for Kadomstev-Petviashvili solutions in a warm dusty plasma with opposite polarity

Kadomstev-Petviashvili (KP) equation is derived using reductive perturbation method. This equation transformed into a nonlinear Schrödinger equation (NLS) by using appropriate variable transformations. When the carrier wave frequency is much smaller than the dust plasma frequency, the DA waves generating modulated wave packets in the form of rogue waves. The dependence of rogue wave profile on system plasma parameters investigated numerically. The parameters in this model are within the ranges corresponding to upper mesosphere, cometary tails and Jupiter’s magnetosphere.     

New Ansaetze for Obtaining Exact Solutions of Nonlinear Schroedinger-Type Equation

We propose two simple ansaetze that allow us to obtain different analytical solutions for two generalizeal versions of the nonlinear Schrodinger equation, such as the averaged dispersive-managed fiber system equation and the extended nonlinear Schrodinger equation which describe the femtosecond pulse propagation in monomode optical fiber. Among these solutions we can find solitary wave and periodic wave solutions representing the propagation of different waveforms in nonlinear media.     

Rogue internal waves in the ocean: Long wave model

Rogue waves can be categorized as unexpectedly large waves, which are temporally and spatially localized. They have recently received much attention in the water wave context, and also been found in nonlinear optical fibers. In this paper, we examine the issue of whether rogue internal waves can be found in the ocean. Because large-amplitude internal waves are commonly observed in the coastal ocean, and are often modeled by weakly nonlinear long wave equations of the Korteweg-de Vries type, we focus our attention on this shallow-water context. Specifically, we examine the occurrence of rogue waves in the Gardner equation, which is an extended version of the Korteweg-de Vries equation with quadratic and cubic nonlinearity, and is commonly used for the modelling of internal solitary waves in the ocean. Importantly, we choose that version of the Gardner equation for which the coefficient of the cubic nonlinear term and the coefficient of the linear dispersive term have the same sign, as this allows for modulational instability. From numerical simulations of the evolution of a modulated narrow-band initial wave field, we identify several scenarios where rogue waves occur.     

Under which conditions the Benjamin-Feir instability may spawn an extreme wave event: A fully nonlinear approach

Within the framework of the fully nonlinear water waves equations, we consider a Stokes wavetrain modulated by the Benjamin-Feir instability in the presence of both viscous dissipation and forcing due to wind. The wind model corresponds to the Miles’ theory. By introducing wind effect on the waves, the present paper extends the previous works of [6] and [7] who neglected wind input. It is also a continuation of the study developed by [9] who considered a similar problem within the framework of the NLS equation. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by [9] from a linear stability analysis. Furthermore, it is found that wind input goes in the subharmonic mode of the modulation whereas dissipation damps the fundamental mode of the initial Stokes wavetrain.     

Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.     

Localized Optical Waves in Defocusing Regime of Negative-Index Materials

We study optical localized waves on a plane-wave background in negative-index materials governed by the defocusing nonlinear Schrodinger equation with self-steepening effect. Important characteristics of localized waves,such as the excitations, transitions, propagation stability, and mechanism, are revealed in detail. An intriguing sequential transition that involves the rogue wave, antidark-dark soliton pair, antidark soliton and antidark soliton pair can be triggered as the self-steepening effect attenuates. The corresponding phase diagram is established in the defocusing regime of negative-index materials. The propagation stability of the localized waves is confirmed numerically. In particular, our results illuminate the transition mechanism by establishing the exact correspondence between the transition and the modulation instability analysis.