Two-dimensional spatial solitons in highly nonlocal nonlinear media

We investigate, analytically and numerically, a class of novel higher-order spatial solitons in two transverse-dimensions, in highly nonlocal nonlinear media. The stability of these solutions in propagation is confirmed by direct numerical simulation. Our results demonstrate that the higher-order spatial solitons in highly nonlocal nonlinear media can exist in various forms, such as the fundamental solitons, vortex-ring solitons, multipole solitons, and fractional solitons.     

Self-Similar Hermite-Gaussian Spatial Solitons in Two-Dimensional Nonlocal Nonlinear Media

We study analytically and numerically the propagation of spatial solitons in a two-dimensional strongly nonlocal nonlinear medium. Exact analytical solutions in the form of self-similar spatial solitons are obtained involving higher-order Hermite-Gaussian functions. Our theoretical predictions provide new insights into the low-energy spatial soliton transmission with high fidelity.     

Sine Hollow Solitons and Breathers in Nonlocal Media

The nonlocal nonlinear Schrödinger equation (NNLSE) describes the propagation dynamics of optical beams in nonlinear media with a spatial nonlocal response. Based on NNLSE, we obtain the generalized sine hollow solitons and breathers and show that the transverse intensity evolutions of them are always periodical. However, if the incident power takes a critical value, the beam width can remain invariant during the propagation, just like the solitons. Otherwise, the beam width varies periodically, just like the breathers. We investigate the evolution characteristics for both cases analytically and numerically in detail.     

Interaction of Nonlocal Incoherent White-Light Solitons

The propagation and interaction of nonlocal incoherent white-light solitons in strongly nonlocal kerr media is investigated. Numerical simulations show that the interaction properties of nonlocal incoherent white-light solitons are different from the case in local media. The interactions of nonlocal incoherent white-light solitons are always attractive independent of their relative phase, while the other parameters such as the extent of nonlocality and the input power have a great impact on the soliton interactions. Pertinent numerical examples are presented to show their propagation and interaction behaviour further.     

Wave propagation in fluid-filled single-walled carbon nanotube on analytically nonlocal Euler–Bernoulli beam model

An analytically nonlocal Euler–Bernoulli beam model for the wave propagation in fluid-filled single-walled carbon nanotube (SWCNT) is established. The governing equations with the nonlocal effects are derived on the variational principle, and used in the wave propagation analysis of the SWCNT beam. Compared with the partially nonlocal Euler–Bernoulli beam models used previously, the analytically nonlocal model presented in the present study predicts well the effects of the stiffness enhancement and the wave damping at the high wavenumber or the strong nonlocal effects area for the fluid-filled SWCNT beam. Though the analytical model is less sensitive than the partially nonlocal model when the moving velocity of the internal fluid is high enough, it simulates more of the high-order nonlocal effecting information than the partially nonlocal model does in many cases.     

强非局域光晶格中空间孤子的脉动传播

根据强非局域结构中空间孤子的演化方程——非局域非线性薛定谔方程,采用分步傅里叶方法,对一维强非局域光晶格结构中空间孤子的脉动传播进行数值研究.分析了孤子的初始光束能量、非局域程度、光晶格调制强度、光晶格周期以及线性折射率的纵向调制率与空间孤子的脉动传播周期之间的关系,并对影响脉动传播的内在物理机制进行了讨论.同时,还研究了在线性折射率纵向调制情况下强非局域光晶格结构中空间孤子传输的开关行为. 关键词： 非局域非线性薛定谔方程 光晶格 空间孤子 光开关     

Spatial nonlocal pair correlations in a repulsive 1D Bose gas

We analytically calculate the spatial nonlocal pair correlation function for an interacting uniform 1D Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects.     

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter ε tends to zero. Assuming natural initial data having the profile of a moving −2π kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all ε>0 sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small ε. This sequence of exact solutions is analogous to that of the well-known N-soliton (or higher-order soliton) solutions of the focusing nonlinear Schrödinger equation. We then use plots obtained from a careful numerical implementation of the inverse-scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit. In the limit ε↓0 one observes the appearance of nonlinear caustics, i.e. curves in space-time that are independent of ε but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases.In the appendices, we give a self-contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic L¹-Sobolev initial data, and Appendix B establishes the well-posedness for L^p-Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).     

New positon, negaton and complexiton solutions for the Hirota-Satsuma coupled KdV system

New positon, negaton and complexiton solutions for the Hirota-Satsuma coupled KdV system are constructed by means of the Darboux transformation with zero seed solution. The new positon, negaton and complexiton solutions are singular and given out both analytically and graphically.     

Solitons in optical lattices with nonlocal nonlinearity

The propagation of spatial solitons is systematically investigated in nonlocal nonlinear media with an imprinted transverse periodic modulation of the refractive index. Based on the variational principle and the infinitesimal approximation of Maclaurin series expansion, we obtain an analytical solution of such nonlocal spatial solitons and an interesting result that the critical power for such solitons propagation is smaller than that in uniform nonlocal self-focusing media. It is found that there exist thresholds in modulation period and lattice depth for such solitons. A stable spatial soliton propagation is maintained with proper adjustment of the modulation period and the lattice depth.     

New positon, negaton and complexiton solutions for the Bogoyavlensky-Konoplechenko equation

New positon, negaton and complexiton solutions for the Bogoyavlensky-Konoplechenko equation are constructed by means of the Darboux transformation with constant seed solution. The new positon, negaton and complexiton solutions are analytical or singular and given out both analytically and graphically.     

Investigating Bulk Waves in Orthotropic Rectangular Nanoplates Based on Three Dimensional Elasticity Theory and Nonlocal Elasticity Theory

The propagation of bulk waves in rectangular nanoplates is studied on the basis of nonlocal three-dimensional elasticity theory. The nonlocal theory applies to both thin and thick rectangular orthotropic nanoplates. The dispersion relation for the waves is derived analytically. Our results are checked against data for macroplates. The influence of nonlocality and other parameters on the wave frequency and phase velocity is discussed.     

Geometric phase and non-stationary state

By investigating a particle motion in a three-dimensional potential barrier with moving boundary, we find that due to an alteration of boundary conditions, the wave function pick up an additional nonlocal phase factor independent on the dynamics of physical system. By compare the nonlocal phase with the geometric phase of the physical system, furthermore, we find that the nonlocal feature of quantum behavior can fully be described by its geometric phase.     

非局域非线性介质中空间暗孤子的理论和实验研究

对非局域非线性介质中的空间暗孤子进行了研究.理论上运用牛顿迭代法求解非局域非线性薛定谔方程,得到了不同传播常数下的非局域空间暗孤子的数值解,发现在任何非局域程度以及任何传播常数条件下,都存在暗孤子的解,而且孤子的束宽与非局域程度存在一定的关系.实验上,在染料溶液中观测到了空间暗孤子在非局域非线性介质中的形成.利用输入功率所引起的非线性效应强度的变化,分析了背景光波形对暗孤子的影响,数值模拟结果与实验结果相符合. 关键词： 非局域非线性 空间暗孤子     

Dipole solitons in nonlocal nonlinear media with anisotropy

We investigate the existence and stability of dipole-mode solitons in two-dimensional models of nonlocal media with anisotropic Kerr nonlinearity analytically and numerically. We obtain the approximate solution of such elliptic dipole solitons by using the variational approximation. The dynamics of such dipole-mode solitons is governed by the eccentricity of both the input beam and the nonlocal response function. We also compute the stability of the solitons by direct numerical simulations. The effects of the anisotropy of the nonlocal response function on the propagation of the dipole beam are also discussed in detail.     

Growth,spectral properties,and laser demonstration of Nd:GYSO crystal

A generic nonlocal nonlinear optical system with a diffusive type of nonlinearity is investigated analytically, using the homogeneous balance principle and the F-expansion technique. Exact traveling wave and soliton solutions are discovered. Numerical simulation of their propagation and interaction properties is carried out. Our results demonstrate that the nonlocal solitary waves can be manipulated and controlled by changing the nonlocality parameter.     

Exact analytic self-similar solution of a wave attractor field

Stratified and rotating fluids support obliquely propagating internal waves. A symmetry-breaking shape of the fluid domain focuses them on a wave attractor. For a trapezoidal basin, it is here shown how to determine the internal wave field analytically. This requires solving the wave equation on a closed domain-an ill-posed Cauchy problem-whose solution exhibits a remarkable self-similar spatial structure. These results are relevant for mixing and mean flow generation in oceans, atmospheres and stars whose symmetry is generally broken and where internal waves are tidally forced.     

Wave propagation in double-walled carbon nanotubes on a novel analytically nonlocal Timoshenko-beam model

This paper is concerned with the characteristics of wave propagation in double-walled carbon nanotubes (DWCNTs). The DWCNTs is simulated with a Timoshenko beam model based on the nonlocal continuum elasticity theory, referred to as an analytically nonlocal Timoshenko-beam (ANT) model. The governing equations of the DWCNTs beam consist of a set of four equations that are derived from the variational principle of the beam with high-order boundary conditions at the both ends, in which the effects of the nano-scale nonlocality and the van der Waals interaction between inner and outer tubes are inclusive. The characteristics of the wave propagation in the DWCNTs beam were analyzed with the new ANT model proposed and the comparisons with the partially nonlocal Timoshenko-beam (PNT) models in publication were made in details. The results show that the nonlocal effects of the ANT model proposed in the present study on the wave propagations are more significant because it is in stronger stiffness enhancement to the DWCNTs beam.     

双曲正割光束在弱非局域非线性体材料中的传输特性研究

利用变分法研究了(2+1)维圆对称双曲正割光束在弱非局域非线性介质中的传输,得到了描述光束束宽、相位、波前曲率、振幅演化的一组微分方程,并得到了光束做孤子传输的临界功率;通过稳定性分析给出了弱非局域情形非局域效应对光束传输的稳定作用的定量描述,从而自洽地阐述了由不稳定的(2+1)维克尔孤子到稳定的(2+1)维弱非局域孤子的过渡情形. 数值模拟的结果验证了变分计算结果的正确性,并说明圆对称的双曲正割函数是(2+1)维弱非局域空间孤子的很好的近似. 关键词： 双曲正割光束 弱非局域非线性介质 空间光孤子     

Numerical computation of the asymptotic size of the rotation domain for the Arnold family

We consider the Arnold Tongue of the Arnold family of circle maps associated to a fixed Diophantine rotation number θ. The corresponding maps of the family are analytically conjugate to a rigid rotation. This conjugation is defined on a (maximal) complex strip of the circle and, after a suitable scaling, the size of this strip is given by an analytic function of the perturbative parameter.The main purpose of this paper is to perform a numerical accurate computation of this function and of its Taylor expansion. This allows us to verify previous theoretical results. The rotation numbers we select are quadratic irrationals, mainly the Golden Mean.By introducing a nonstandard extrapolation process, especially suited for the problem, we compute all the quantities required (rotation numbers, Arnold Tongues, Fourier and Taylor coefficients) with high precision.