强非局域非线性介质中光束传输的厄米高斯解

利用强非局域非线性介质中空间对称实响应函数的泰勒展开简化了非局域非线性薛定谔方程 ，文章基于强非局域非线性空间中的线性模型得到了矩空间1+D(D=1，2)维的厄米高斯 型解，得到了高阶孤子波的解析式，高斯解是最低阶孤子，即基模光孤子，并得到了入射光 束的临界功率．图示展现了几个低阶光孤子解，并发现了强非局域非线性介质中存在非对称 光孤子． 关键词： 高阶孤子 强非局域介质 厄米高斯     

非局域克尔介质中厄米高斯光束传输的变分研究

在非局域非线性克尔介质中,通过对介质实对称响应函数的泰勒展开,简化了非局域非线性薛定谔方程所对应的Lagrange密度,进而利用变分法对光束的传输问题进行了分析.求出试探解各个参量的演化方程并得到了自聚焦介质中的厄米高斯型光束的精确解析解,当输入功率达到临界功率时,即形成高阶空间光孤子(厄米高斯孤子),其最低阶(基模光孤子)就是高斯孤子.通过数值模拟发现解析解与数值解符合得很好. 关键词： 非局域克尔介质 变分法 厄米高斯光束 空间光孤子     

椭圆强非局域空间光孤子

对傍轴椭圆高斯光束在具有椭圆对称响应特性的强非局域非线性介质中的演化规律进行研究，得到了光束各参量演化的精确解析解，分析了单向空间光孤子和强非局域椭圆空间光孤子的形成条件，发现了椭圆光孤子的相移与介质响应函数的椭圆率有关. 关键词： 椭圆对称强非局域响应介质 椭圆强非局域空间光孤子 相移     

(1+2)维空间光孤子在非局域克尔介质中的传输特性

 从非局域非线性薛定谔方程出发，采用分步傅里叶算法数值讨论了在一定的非局域程度条件下，（1+2）维空间光孤子的传输特性， 数值求解了光孤子各特性参量。假定非局域克尔介质的响应函数为高斯型，得出了在一定的非局域程度条件下空间光孤子的数值解，并数值证明了它们的稳定性。结果表明:（1+2）维光孤子对非局域程度依赖性很强。在一定的非局域程度下，光束能以光孤子态在非局域克尔介质中稳定传输。强非局域时，光孤子的波形是高斯型，其它的非局域程度下，不是高斯型。当非局域程度较弱时，不存在孤子解。     

(1+2)维各向同性介质中的旋转椭圆空间光孤子

从(1+2)维非局域非线性薛定谔方程出发, 通过坐标变换得到了旋转坐标系下的非局域非线性薛定谔方程. 假设响应函数为高斯型, 用虚时间法数值求解了旋转坐标系下的非局域非线性薛定谔方程的静态孤子解, 迭代出了不同非局域程度条件下的静态椭圆孤子数值解. 最后采用分步傅里叶算法, 以迭代的孤子解作为初始输入波形, 模拟了在不同的非局域程度条件下, (1+2)维椭圆空间光孤子的旋转传输特性. 强非局域时, 椭圆光孤子的长轴方向和短轴方向波形都是高斯型, 其他的非局域程度下, 不是高斯型. 由此表明：(1+2)维椭圆光孤子对非局域程度依赖性很强. 旋转角速度和功率均与非局域程度以及孤子的椭圆度有关.     

椭圆高斯光束在强非局域非线性介质中的传输特性

研究了傍轴椭圆高斯光束在强非局域非线性介质中的传输特性，得到了其各参量的演化方程 及其精确解析解.通过对束宽演化方程及其精确解析解的进一步分析，发现傍轴椭圆高斯光 束在强非局域非线性介质中传输时，两横向方向束宽作周期性变化.不管初始功率为多大， 光束都将周期性的由椭圆高斯光束演化为圆对称高斯光束，再由圆对称高斯光束演化为椭圆 高斯光束；并且在演化的过程中，椭圆的半长轴和半短轴会作周期性交替变化.另外，在一 定初始功率下，傍轴椭圆高斯光束可以保持某一横向方向的束宽不变，得到光孤子. 关键词： 强非局域非线性介质 非局域非线性薛定谔方程 椭圆高斯光束 参量演化方程 空间孤子     

不同非局域程度条件下空间光孤子的传输特性

光束在非局域非线性介质中传输由非局域非线性薛定谔方程描述.讨论了在不同非局域程度 条件下，空间光孤子的传输特性.提出了一个基于分步傅里叶算法数值求解孤子波形和分布 的迭代算法.假定介质的非线性响应函数为高斯型，得出了在不同非局域程度条件下空间光 孤子的数值解，并数值证明了它们的稳定性.结果表明，不论非局域程度如何，光束都能以 光孤子态在介质中稳定传输.光孤子的波形是从强非局域时的高斯型过渡到局域时的双曲正 割型，形成孤子的临界功率随非局域程度的减弱而减小，光孤子相位随距离线性增大，相位 的变化率随非局域程度的减弱而减小. 关键词： 非局域非线性薛定谔方程 空间光孤子 临界功率 相位     

向列相液晶中强非局域空间光孤子传输的理论研究

对向列相液晶中非局域空间孤子的传输进行了理论研究.基于非线性液晶孤子传输方程，采用Gauss形式的试探解，不仅得到了空间孤子的解析解,而且还在临界功率附近得到了呼吸子的解析解.通过数值模拟证明我们的结果比Conti和Assanto等人的结果更合理.同时，对液晶中的非局域孤子模型和Snyder等提出的强非局域孤子模型进行了全面的比较. 关键词： 向列相液晶 空间孤子 非局域非线性 呼吸子     

离散的五次非线性薛定谔方程的精确孤子解

利用推广的双曲函数法，我们研究了离散的五次非线性薛定谔方程。当方程的系数满足某种约束关系时，我们得到了新的局域解。这些解包括亮孤子解，暗孤子解，相位交替变化的亮孤子解和相位交替变化的暗孤子解。     

强非局域非线性介质中的旋转涡旋光孤子

利用变分方法,得到了Snyder-Mitchell模型的旋转空间调制涡旋光孤子的近似解析解.在传输过程中,这种光孤子具有可观察的旋转特性.在一定的条件下,旋转的空间调制涡旋光孤子将退化为圆对称的涡旋光孤子. 关键词： 非局域非线性介质 强非局域性 空间调制涡旋光孤子 变分法     

Exact Solutions to (2+1)-Dimensional Kaup--Kupershmidt Equation

In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the (2+1)-dimensional Kaup-Kupershmidt (KK) equation are presented. We successfully obtain more general exact travelling wave solutions for (2+1)-dimensional KK equation by the symmetry method and the (G^, /G)-expansion 　method. Consequently, we find some new solutions of (2+1)-dimensional KK equation, 　including similarity solutions, solitary wave solutions, and 　periodic solutions.     

Symmetry Reduction and Exact Solutions of the (3+1)-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the (3+1)-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the (3+1)-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the (3+1)-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the (3+1)-dimensional breaking soliton equation.     

Topological structure of the solitons solution in SU(3) Dunne-Jackiw-Pi-Trugenberger model

By using φ-mapping topological current theory and gauge potential decomposition, we discuss the self-dual equation and its solution in the SU(N) Dunne-Jackiw-Pi-Trugenberger model and obtain a new concrete self-dual equation with a δ function. For the SU(3) case, we obtain a new self-duality solution and find the relationship between the soliton solution and topological number which is determined by the Hopf index and Brouwer degree of φ-mapping. In our solution, the flux of this soliton is naturally quantized.     

Soliton solutions,travelling wave solutions and conserved quantities for a three-dimensional soliton equation in plasma physics

Many physical systems can be successfully modelled using equations that admit the soliton solutions. In addition, equations with soliton solutions have a significant mathematical structure. In this paper, we study and analyze a three-dimensional soliton equation, which has applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories. Indeed, solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour. We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time. Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function, elliptic functions, elementary trigonometric and hyperbolic functions solutions of the equation. Besides, various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique. These solutions comprise dark soliton, doubly-periodic soliton, trigonometric soliton, explosive/blowup and singular solitons. We further exhibit the dynamics of the solutions with pictorial representations and discuss them. In conclusion, we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula. We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new.     

A Bilinear Backlund Transformation and Explicit Solutions for a （3＋1）-Dimensional Soliton Equation

Considering the bilinear form of a （3＋1）-dimensional soliton equation, we obtain a bilinear Backlund transformation for the equation. As an application, soliton solution and stationary rational solution for the （3＋1）- dimensional soliton equation are presented.     

Soliton dynamics and elastic collisions in a spin chain with an external time-dependent magnetic field

In this paper, we investigate the nonlinear dynamics of a Heisenberg spin chain with an external time-oscillating magnetic field. Such dynamics can be described by a Landau–Lifshitz-type equation. We apply the Darboux transformation method to the linear eigenvalue problem associated with this equation, and obtain the multi-soliton solution with a purely algebraic iterative procedure. Through the analytical analysis and graphical illustrations for the solutions obtained, we discuss in detail the effects of an external magnetic field on the nonlinear wave. Under the action of an external field, although the amplitude, width and depth of soliton vary periodically with time and its symmetry property is changeable, the soliton can also propagate stably and it possesses particle-like behavior.     

Fifth-Order Alice-Bob Systems and Their Abundant Periodic and Solitary Wave Solutions*

The study on the nonlocal systems is one of the hot topics in nonlinear science. In this paper, the well-known fifth-order integrable systems including the Sawada-Kotera (SK) equation, the Kaup-Kupershmidt (KK) equation and the fifth-order Koterweg-de Vrise (FOKdV) equation are extended to a generalized two-place nonlocal form, the generalised fifth-order Alice-Bob system. The Lax integrability of two sets of Alice-Bob systems for all the SK, KK and FOKdV type systems are explicitly given via matrix Lax pairs. The $\hat{P}\hat{T}$ symmetry breaking and symmetry invariant periodic and solitary waves for one set of nonlocal SK, KK and FOKdV system are investigated via a special travelling wave solution ansatz.     

(3+1)维Burgers系统的瞬内嵌孤子和瞬锥形孤子

利用投射方程法和变量分离法,得到了(3+1)维Burgers系统的变量分离解(包括孤波解、周期波解和有理函数解).根据孤波解和有理函数解,构造出Burgers系统新颖的局域结构,例如瞬内嵌孤子和瞬锥形孤子.     

On symmetry preserving and symmetry broken bright,dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation

We construct symmetry preserving and symmetry broken N-bright, dark and antidark soliton solutions of a nonlocal nonlinear Schrödinger equation. To obtain these solutions, we use appropriate eigenfunctions in Darboux transformation (DT) method. We present explicit one and two bright soliton solutions and show that they exhibit stable structures only when we combine the field and parity transformed complex conjugate field. Further, we derive two dark/antidark soliton solution with the help of DT method. Unlike the bright soliton case, dark/antidark soliton solution exhibits stable structure for the field and the parity transformed conjugate field separately. In the dark/antidark soliton solution case we observe a contrasting behaviour between the envelope of the field and parity transformed complex conjugate envelope of the field. For a particular parametric choice, we get dark (antidark) soliton for the field while the parity transformed complex conjugate field exhibits antidark (dark) soliton. Due to this surprising result, both the field and PT transformed complex conjugate field exhibit sixteen different combinations of collision scenario. We classify the parametric regions of dark and antidark solitons in both the field and parity transformed complex conjugate field by carrying out relevant asymptotic analysis. Further we present 2N-dark/antidark soliton solution formula and demonstrate that this solution may have $2^{2N}\times 2^{2N}$ combinations of collisions.     

Nonlocal Optical Spatial Soliton with a Non-parabolic Symmetry and Real-valued Convolution Response Kernel

Based on the picture of nonlinear and non-parabolic symmetry response, i.e., Δn₂(I)≈ρ(a₀+a₁x-a₂x²), we propose a model for the transversal beam intensity distribution of the nonlocal spatial soliton. In this model, as a convolution response with non-parabolic symmetry, Δ n₂(I)≈ρ(b₀+b₁f-b₂f² with b₂/b₁>0 is assumed. Furthermore, instead of the wave function Ψ, the high-order nonlinear equation for the beam intensity distribution f has been derived and the bell-shaped soliton solution with the envelope form has been obtained. The results demonstrate that, since the existence of the terms of non-parabolic response, the nonlocal spatial soliton has the bistable state solution. If the frequency shift of wave number β satisfies 0<4(β-ρb₀/μ)<3η₀/8α, the bistable state soliton solution is stable against perturbation. It should be emphasized that the soliton solution arising from a parabolic-symmetry response kernel is trivial. The sufficient condition for the existence of bistable state soliton solution b₂/b₁>0 has been demonstrated.