Conformal structure-preserving method for damped nonlinear Schrdinger equation

In this paper,we propose a conformal momentum-preserving method to solve a damped nonlinear Schrodinger(DNLS) equation.Based on its damped multi-symplectic formulation,the DNLS system can be split into a Hamiltonian part and a dissipative part.For the Hamiltonian part,the average vector field(AVF) method and implicit midpoint method are employed in spatial and temporal discretizations,respectively.For the dissipative part,we can solve it exactly.The proposed method conserves the conformal momentum conservation law in any local time-space region.With periodic boundary conditions,this method also preserves the total conformal momentum and the dissipation rate of momentum exactly.Numerical experiments are presented to demonstrate the conservative properties of the proposed method.     

变系数非线性Schrödinger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Schrödinger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Schrödinger方程转化成非线性Schrödinger方程,并利用已知解深入研究变系数非线性Schrödinger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. 关键词： 非线性Schrö dinger方程 相似变换 变系数 孤子解     

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper,we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear Schrdinger equation (NLS) with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions,one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results,some previous one-and two-soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one-and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.     

Chaos control in the nonlinear Schrdinger equation with Kerr law nonlinearity

The nonlinear Schrdinger equation with Kerr law nonlinearity in the two-frequency interference is studied by the numerical method.Chaos occurs easily due to the absence of damping.This phenomenon will cause the distortion in the process of information transmission.We find that fiber-optic transmit signals still present chaotic phenomena if the control intensity is smaller.With the increase of intensity,the fiber-optic signal can stay in a stable state in some regions.When the strength is suppressed to a certain value,an unstable phenomenon of the fiber-optic signal occurs.Moreover we discuss the sensitivities of the parameters to be controlled.The results show that the linear term coefficient and the environment of two quite different frequences have less effects on the fiber-optic transmission.Meanwhile the phenomena of vibration,attenuation and escape occur in some regions.     

Data-driven parity-time-symmetric vector rogue wave solutions of multi-component nonlinear Schrödinger equation

Rogue waves are a class of nonlinear waves with extreme amplitudes, which usually appear suddenly and disappear without any trace. Recently, the parity-time ($\mathcal {PT}$)-symmetric vector rogue waves (RWs) of multi-component nonlinear Schrödinger equation ($n$-NLSE) are usually derived by the methods of integrable systems. In this paper, we utilize the multi-stage physics-informed neural networks (MS-PINNs) algorithm to derive the data-driven $\mathcal {PT}$ symmetric vector RWs solution of coupled NLS system in elliptic and X-shapes domains with nonzero boundary condition. The results of the experiment show that the multi-stage physics-informed neural networks are quite feasible and effective for multi-component nonlinear physical systems in the above domains and boundary conditions.     

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     

Nonautonomous solitons in the continuous wave background of the variable-coefficient higher-order nonlinear Schro¨dinger equation

We reduce the variable-coefficient higher-order nonlinear Schrdinger equation (VCHNLSE) into the constantcoefficient (CC) one. Based on the reduction transformation and solutions of CCHNLSE, we obtain analytical soliton solutions embedded in the continuous wave background for the VCHNLSE. Then the excitation in advancement and sustainment of soliton arrays, and postponed disappearance and sustainment of the bright soliton embedded in the background are discussed in an exponential system.     

Solving coupled nonlinear Schrödinger equations via a direct discontinuous Galerkin method

In this work, we present the direct discontinuous Galerkin (DDG) method for the one-dimensional coupled nonlinear Schrödinger (CNLS) equation. We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system. The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method. Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.     

Soliton,breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints

The nonlinear Schro¨dinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schro¨dinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schro¨dinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.     

Multi-symplectic variational integrators for nonlinear Schrdinger equations with variable coefficients

In this paper, we propose a variational integrator for nonlinear Schrdinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrdinger equations with variable coefficients, cubic nonlinear Schrdinger equations and Gross–Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.     

Collapse arrest in the space-fractional Schrödinger equation with an optical lattice

The soliton solution and collapse arrest are investigated in the one-dimensional space-fractional Schr?dinger equation with Kerr nonlinearity and optical lattice. The approximate analytical soliton solutions are obtained based on the variational approach, which provides reasonable accuracy. Linear-stability analysis shows that all the solitons are linearly stable. No collapses are found when the Lévy index 1 α≤ 2. For α = 1, the collapse is arrested by the lattice potential when the amplitude of perturbations is small enough. It is numerically proved that the energy criterion of collapse suppression in the two-dimensional traditional Schr?dinger equation still holds in the one-dimensional fractional Schr?dinger equation. The physical mechanism for collapse prohibition is also given.     

Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schrödinger equation

We make a quantitative study on the soliton interactions in the nonlinear Schrödinger equation (NLSE) and its variable-coefficient (vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities (especially the soliton accelerations and interaction forces); whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles, particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.     

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

Exact projective solutions of generalized nonlinear Schrödinger system with variable parameters

A direct self-similarity mapping approach is successfully applied to a generalized nonlinear Schrödinger (NLS) system. Based on the known exact solutions of a self-similarity mapping equation, a few types of significant localized excitation with novel properties are obtained by selecting appropriate system parameters. The integrable constraint condition for the generalized NLS system derived naturally here is consistent with the known compatibility condition generated via the Painlev? analysis.     

Analytical solutions and rogue waves in (3+1)-dimensional nonlinear Schrödinger equation

Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrödinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.     

耦合Schrödinger系统的周期振荡折叠孤子

引入对称延拓和非线性变换, 将(G'/G)展开法扩展到研究(1+1)维非线性耦合Schrödinger系统, 构造出该系统的一些分离变量形式的精确解. 通过对解中的任意函数进行适当的设置, 获得了两类周期振荡折叠孤子. 关键词： 耦合Schrö dinger系统 G'/G)展开法')" href="#">(G'/G)展开法 精确解 周期振荡折叠孤子     

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrdinger equation with variable coefficients

In this paper, the extended symmetry transformation of (3+1)-dimensional (3D) generalized nonlinear Schrdinger (NLS) equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.     

Dynamics of solitons of the generalized (3+1)-dimensional nonlinear Schrödinger equation with distributed coefficients

We present three families of soliton solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation with distributed coefficients. We investigate the dynamics of these solitons in nonlinear optics with some selected parameters. Different shapes of bright solitons, a train of bright solitons and dark solitons are observed. The obtained results may raise the possibilities of relevant experiments and potential applications.     

Propagations of Fresnel diffraction accelerating beam in Schrödinger equation with nonlocal nonlinearity

Accelerating beams have been the subject of extensive research in the last few decades because of their self-acceleration and diffraction-free propagation over several Rayleigh lengths. Here, we investigate the propagation dynamics of a Fresnel diffraction beam using the nonlocal nonlinear Schrödinger equation (NNLSE). When a nonlocal nonlinearity is introduced into the linear Schrödinger equation without invoking an external potential, the evolution behaviors of incident Fresnel diffraction beams are modulated regularly, and certain novel phenomena are observed. We show through numerical calculations, under varying degrees of nonlocality, that nonlocality significantly affects the evolution of Fresnel diffraction beams. Further, we briefly discuss the two-dimensional case as the equivalent of the product of two one-dimensional cases. At a critical point, the Airy-like intensity profile oscillates between the first and third quadrants, and the process repeats during propagation to yield an unusual oscillation. Our results are expected to contribute to the understanding of NNLSE and nonlinear optics.