立方非线性Schr?dinger方程的动力学性质研究及其解模式的漂移

采用辛算法数值求解一维立方非线性Schr?dinger方程，研究了随着非线性参数的变化立方非线性Schr?dinger方程的动力学性质和解的模式的漂移.数值结果表明，随着非线性参数的增加解模式的漂移速度越来越快. 关键词： 动力学性质 相轨线 解模式的漂移 辛算法     

非线性Schrdinger方程的动力学行为分析

采用辛算法数值求解非线性Schr dinger方程的周期初值问题,建立不同的相空间来分析其动力学特性.首先比较分析了不同的相空间中立方非线性Schr dinger方程在不同立方非线性参数下的长时间演化的动力学特性,然后讨论了相空间中立方-五次方非线性Schr dinger方程在不同立方和五次方非线性参数下的长时间演化的动力学特性,数值结果显示,对于不同的立方非线性参数,随着五次方非线性参数的增加,动力学行为的演化路径是不一样的.     

立方五次方非线性Schrdinger方程的动力学性质研究

采用辛算法数值求解了一维立方五次方非线性Schrdinger方程,研究了不同非线性参数下非线性Schrdinger方程的动力学性质.数值结果表明,随着立方非线性参数的增加,系统经历了拟周期状态、混沌状态和周期状态,且在五次方项的调制下,呼吸子解可以退化为单孤子解.     

变系数非线性Schr(o)dinger方程的孤子解及其相互作用

在光孤子通信和Bose-Einstein凝聚体动力学研究中,求解广义非线性Schr(o)dinger方程是一个重要的研究方向.稳定的孤子模式具有潜在的应用,可为实验技术的实现提供依据.本文引进一种相似变换,将变系数非线性Schr(o)dinger方程转化成非线性Schr(o)dinger方程,并利用已知解深入研究变系数非线性Schr(o)dinger方程解的单孤子解、两孤子解和连续波背景下的孤子解.同时通过选择不同的具体参数,给出它们的图像分析和相应的讨论.     

光纤中变系数非线性Schrödinger方程的孤子解及其应用

基于推广的立方非线性Klein-Gordon方程对一般形式的变系数非线性Schr(o)dinger方程进行研究,讨论了无啁啾情形的孤子解,发现了包括亮、暗孤子解和类孤子解在内的一些新的精确解. 同时对基本孤子的色散控制方法进行了简单讨论. 作为特例,常系数非线性Schr(o)dinger方程和两类特殊的变系数非线性Schr(o)dinger方程的结果和已知的形式一致.此外,还研究了一个周期增益或损耗的光纤系统,得到了有意义的结果.     

A Direct Discontinuous Galerkin Method for Nonlinear Schr(o)dinger Equation

讨论一维和二维非线性Schr(o)dinger (NLS)方程的数值求解.基于扩散广义黎曼问题的数值流量,构造一种直接间断Galerkin方法(DDG)求解非线性Schr(o)dinger方程.证明该方法L2稳定性,并说明DDG格式是一种守恒的数值格式.对一维NLS方程的计算表明,DDG格式能够模拟各种孤立子形态,而且可以保持长时间的高精度.二维NLS方程的数值结果显示该方法的高精度和捕捉大梯度的能力.     

立方五次方非线性Schrodinger方程的动力学性质研究

采用辛算法数值求解了一维立方五次方非线性Schrödinger方程,研究了不同非线性参数下非线性Schrödinger方程的动力学性质.数值结果表明,随着立方非线性参数的增加,系统经历了拟周期状态、混沌状态和周期状态,且在五次方项的调制下,呼吸子解可以退化为单孤子解. 关键词： 非线性Schrödinger方程 动力学性质 孤子 辛算法     

用Coulomb-Volkov方法研究强激光场中H原子的电离

利用Coulomb-Volkov方法研究了H原子在不同波长的线性极化强激光场中电离的能量谱和动量谱,并与强场近似和直接数值求解含时Schr(o|¨)dinger方程的结果进行比较,结果发现:随着激光频率的增加,由Coulomb-Volkov方法得到的结果与数值求解含时Schr(o|¨)dinger方程的结果符合得很好.     

具有色散系数的(2+1)维非线性Schrdinger方程的有理解和空间孤子

非线性Schrdinger方程是物理学中具有广泛应用的非线性模型之一.本文采用相似变换,将具有色散系数的(2+1)维非线性Schrdinger方程简化成熟知的Schrdinger方程,进而得到原方程的有理解和一些空间孤子.     

变系数非线性Schrdinger方程的孤子解及其相互作用

在光孤子通信和Bose-Einstein凝聚体动力学研究中,求解广义非线性Schrdinger方程是一个重要的研究方向.稳定的孤子模式具有潜在的应用,可为实验技术的实现提供依据.本文引进一种相似变换,将变系数非线性Schrdinger方程转化成非线性Schrdinger方程,并利用已知解深入研究变系数非线性Schrdinger方程解的单孤子解、两孤子解和连续波背景下的孤子解.同时通过选择不同的具体参数,给出它们的图像分析和相应的讨论.     

Polarization nonlinear optics of quadratically nonlinear azopolymers

The polarization properties of second harmonic and sum-frequency signals generated by femtosecond laser pulses in films of polymers containing covalent groups of an azobenzothiazole chromophore polarized by an external electric field are investigated. It is shown that the methods of polarization nonlinear optics make it possible to determine the structure of oriented molecular dipoles and reveal important properties of the motion of collectivized π electrons in organic molecules with strong optical nonlinearities. The polarization measurements show that the tensor of quadratic nonlinear optical susceptibility of chromophore fragments oriented by an external field in macromolecules of the noted azopolymers has a degenerate form. This is indicative of a predominantly one-dimensional character of motion of collectivized π electrons along an extended group of atoms in such molecules.     

Propagation of nonlinear waves in bi-inductance nonlinear transmission lines

We consider a one-dimensional modified complex Ginzburg-Landau equation, which governsthe dynamics of matter waves propagating in a discrete bi-inductance nonlineartransmission line containing a finite number of cells. Employing an extended Jacobielliptic functions expansion method, we present new exact analytical solutions whichdescribe the propagation of periodic and solitary waves in the considered network.     

Exact solutions of nonlinear PDE,nonlinear transformations and reduction of nonlinear PDE to a quadrature

A method to construct the exact solution of the PDE is presents, which combines the two kind methods — the nonlinear transformation and RQ (reduction of the PDE to a quadrature problem) method. The nonlinear diffusion equation is chosen to illustrate the method and the exact solutions are obtained.     

Dynamic testing of nonlinear vibrating structures using nonlinear normal modes

Modal testing and analysis is well-established for linear systems. The objective of this paper is to progress toward a practical experimental modal analysis (EMA) methodology of nonlinear mechanical structures. In this context, nonlinear normal modes (NNMs) offer a solid theoretical and mathematical tool for interpreting a wide class of nonlinear dynamical phenomena, yet they have a clear and simple conceptual relation to the classical linear normal modes (LNMs). A nonlinear extension of force appropriation techniques is developed in this study in order to isolate one single NNM during the experiments. With the help of time-frequency analysis, the energy dependence of NNM modal curves and their frequencies of oscillation are then extracted from the time series. The proposed methodology is demonstrated using two numerical benchmarks, a two-degree-of-freedom system and a planar cantilever beam with a cubic spring at its free end.     

The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals

The third-order nonlinear susceptibilities and nonlinear refractive indices of KDP, BBO, and LiNbO₃ crystals at wavelengths of 1064 and 532 nm are measured using the Z-scan technique. The measurements are made for different energies of incident radiation and different focusing conditions and crystal lengths. It is found that, as the angle between the light beam and the principal optic axis of a KDP crystal increases, the nonlinearity drops, with its magnitude at a wavelength of 1064 nm being higher than at 532 nm. For a BBO crystal, the nonlinearity dispersion is normal. The mechanisms of nonlinear losses in KDP, BBO, and LiNbO₃ crystals are identified. The values of the nonlinear susceptibilities thus obtained are analyzed and compared with the results of calculations based on an empirical model and with the data of other authors.     

Two-dimensional stationary nonlinear surface solitons in nonlocal nonlinear media

We address two-dimensional surface solitons occurring at the interface between a semi-infinite linear medium and a semi-infinite nonlocal nonlinear medium. We find that there exist stable single and dipole surface solitons. The properties of the surface solitons can be affected by the degree of nonlocality. Interestingly, only when the degree of nonlocality is greater than a critical value, the surface solitons can exist.     

Mapping nonlinear gravity into General Relativity with nonlinear electrodynamics

We show that families of nonlinear gravity theories formulated in a metric-affine approach and coupled to a nonlinear theory of electrodynamics can be mapped into general relativity (GR) coupled to another nonlinear theory of electrodynamics. This allows to generate solutions of the former from those of the latter using purely algebraic transformations. This correspondence is explicitly illustrated with the Eddington-inspired Born–Infeld theory of gravity, for which we consider a family of nonlinear electrodynamics and show that, under the map, preserve their algebraic structure. For the particular case of Maxwell electrodynamics coupled to Born–Infeld gravity we find, via this correspondence, a Born–Infeld-type nonlinear electrodynamics on the GR side. Solving the spherically symmetric electrovacuum case for the latter, we show how the map provides directly the right solutions for the former. This procedure opens a new door to explore astrophysical and cosmological scenarios in nonlinear gravity theories by exploiting the full power of the analytical and numerical methods developed within the framework of GR.     

Localization of nonlinear excitations in curved waveguides and chains of nonlinear oscillators

It is shown that the interplay of curvature and nonlinearity in systems with finite curvature: bent waveguides, curved chains of nonlinear oscillators, etc can lead to the qualitative effects, such as symmetry breaking of the nonlinear excitations and their trapping by the bending. The finite curvature of the waveguide with infinite hard walls (Dirichlet boundary conditions) provides a stabilizing effect on otherwise unstable localized states of repelling nonlinear Schr?dinger excitations. The number of quanta which the curved waveguide can bind monotonically increases when the radius of curvature decreases. In the waveguides with Neumann boundary conditions at the confining walls the curved region might manifest itself as a two-hump potential barrier with interbarrier space acting as a potential valley. A threshold character of the scattering process, i.e. transmission, trapping, or reflection of the moving nonlinear excitation passing through the bending, is demonstrated.     

Stability of nonlinear diffusion

The nonlinear diffusion equation in bounded geometry with time-independent boundary conditions has a uniquely determined stationary solution. We show that this solution is dynamically stable in the sense of Liapunov. Any initial distribution tends to the stationary one as time goes on. It is shown that the application of the Glansdorff-Prigogine stability criterion requires a more elaborate analysis. We develop a variational procedure which has application in a wide range of nonlinear transport problems.