Nonautonomous Dark Solitons and Rogue Waves in a Graded-Index Grating Waveguide

We analytically present a family of nonautonomous dark solitons and rogue waves in a planar graded-index grating waveguide with an additional long-period grating.The dark solitons whose dynamics described by the explicit expressions such as the valley,background and wave central position are investigated.We find that dark soliton's depth and the long-period grating have effects on soliton's wave central position;the gain or loss term affects directly both the background and valley of the soliton.For rogue waves,it is reported that one can modulate the distribution of the light intensity by adjusting the parameters of the long-period grating.Additionally,more rogue waves with different evolution behaviors in this special waveguide are demonstrated clearly.     

Nonautonomous Solitons in the(3+1)-Dimensional Inhomogeneous Cubic-Quintic Nonlinear Medium

We have constructed explicit nonautonomous soliton solutions of the generalized nonlinear Schrdinger equation in the(3+1)-dimensional inhomogeneous cubic-quintic nonlinear medium.The gain parameter has no effects on the motion of the soliton's phase or their velocities,and it affects just the evolution of their peaks.As two examples,we discuss the propagation of nonautonomous solitons in the periodic distributed amplification system and the exponential dispersion decreasing system.Results show that the presence of the chirp not only makes the intensity of solitons weaken more promptly,but also broadens their width.     

SU(2)Chern-Simons涡旋解的拓扑结构

运用规范势分解理论研究了 Dunne Jackiw Pi Trugenberger 模型中的自对偶方程, 得到一个静态的自对偶Chern Simons多涡旋解, 每个涡旋由5个参数描述。 发现了自对偶解与拓扑数之间的关系, 而拓扑数由Brouwer度与Hopf指标确定。 同时, 也研究了该涡旋解的磁通量的拓扑量子化。The self dual equation and its solution in SU(2) Dunne Jackiw Pi Trugenberger model has Been discussed with special ansatz for the Lie algebraic structures of su(2) and gauge potential decomposition. We obtainer a new concrete self dual equation and found the relationship between SU(2) Chern Simons vortices and topological number which is determined by Hopf indices and Brouwer degrees of  mapping. (two positions, one scale, one phase per vortex and one charge of each vortex) m vortices solutions are discribed by using 5m parameters. The quantization of flux is also studied in this case.     

Exact Periodic-Wave Solutions for (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional KP Equation

The (2 1)-dimensional Boussinesq equation and (3 1)-dimensional KP equation are studied by using the extended Jacobi elliptic-function method. The exact periodic-wave solutions for the two equations are obtained.     

Exact Periodic-Wave Solutions for （2＋1）-Dimensional Boussinesq Equation and（3＋1）-Dimensional KP Equation

The (2 1)-dimensional Boussinesq equation and (3 1)-dimensional KP equation are studied by using the extended Jacobi elliptic-function method. The exact periodic-wave solutions for the two equations are obtained.     

(2 + 1)-Dimensional Solutions in F(R) Gravity

Motivated by the well-known charged BTZ black holes, we look for (2 + 1)-dimensional solutions of F(R) gravity. At first we investigate some near horizon solutions and after that we obtain asymptotically Lifshitz black hole solutions. Finally, we discuss about rotating black holes with exponential form of F(R) theory.     

A (2+1)-Dimensional Integrable sinh-Gordon Extension

In this letter, we point out that if there is a (2+1)-dimensional extension of the KdV equation,then we can get a corresponding (2+1)-dimensional sine-Gordon (SG) or sinh-Gordon (SHG) extension which is related to the negative flow equation of the KdV extension. The (2+1)-dimensional SG (or SHG) extensions related to the KP, breaking soliton and Nizhnik-Novikov-Veselov equations are known in literature while the (2+1)-dimensional SHG extension related to the negative Aow equation of the Boiti-Leon-Manna-Pempinelli (BLMP) equation is obtained in this letter thanks to the Schwartz form of the BLMP equation being conformal invariant.     

(1+1)-Dimensional Turbulent and Chaotic Systems Reduced from (2+1)-Dimensional Lax Integrable Dispersive Long Wave Equation

After generalizing the Clarkson-Kruskal direct similarity reduction ansatz, one can obtain various newtypes of reduction equations. Especially, some lower-dimensional turbulent systems or chaotic systems may be obtainedfrom the general form of the similarity reductions of a higher-dimensional Lax integrable model. Furthermore, anarbitrary three-order quasi-linear equation, which includes the Korteweg de-Vries Burgers equation and the generalLorenz equation as two special cases, has been obtained from the reductions of the (2+1)-dimensional dispersive longwave equation system. Some types of periodic and chaotic solutions of the system are also discussed.     

Symmetry Reduction of (2+1)-Dimensional Lax–Kadomtsev–Petviashvili Equation

The Lax–Kadomtsev–Petviashvili equation is derived from the Lax fifth order equation, which is an important mathematical model in fluid physics and quantum field theory. Symmetry reductions of the Lax–Kadomtsev–Petviashvili equation are studied by the means of the Clarkson–Kruskal direct method and the corresponding reduction equations are solved directly with arbitrary constants and functions.     

Study of （2＋1）-Dimensional Higher-Order Broer-Kaup System

Painleve property and infinite symmetries of the （2＋1）-dimensional higher-order Broer-Kaup （HBK） system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.ups to （2＋1）-dimensional HBK system. Based on our theorem, some new forms of solutions are obtained. We also find infinite number of conservation laws of the （2＋1）-dimensional HBK system.     

Ti∶LiNbO_3二维渐变折射率光波导传播常数的计算

从Ti∶LiNbO3波导截面折射率分布的特点出发，得出一套计算传播常数简单而精确的方法。文中对该法进行了分析，并应用该方法计算了钛扩散波导的色散关系。该方法为钛扩散光波导集成光学器件的设计提供了一种较可靠的理论手段。     

Oscillating Solitons for (2+1)-Dimensional Nonlinear Models

Using extended homogeneous balance method and variable separation hypothesis,we found new variableseparation solutions with three arbitrary functions of the (2 1)-dimensional dispersive long-wave equations.Based on derived solutions,we revealed abundant oscillating solitons such as dromion,multi-dromion,solitoff,solitary waves,and so on,by selecting appropriate functions.     

Lump Solution of (2+1)-Dimensional Boussinesq Equation

A class of lump solutions of(2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method.Some contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero.The particular lump solutions with specific values of the involved parameters are plotted,as illustrative examples.     

Exact Solutions of (2+1)-Dimensional HNLS Equation

In this paper, we use the classical Lie group symmetry method to get the Lie point symmetries of the (2+1)-dimensional hyperbolic nonlinear Schrödinger (HNLS) equation and reduce the (2+1)-dimensional HNLS equation to some (1+1)-dimensional partial differential systems. Finally, many exact travelling solutions of the (2+1)-dimensional HNLS equation are obtained by the classical Lie symmetry reduced method.     

Nonpropagating Solitary Waves in (2+1)-Dimensional Nonlinear Systems

By means of extended homogeneous balance method and variable separation approach, quite a general variable separation solution of the (2+1)-dimensional Broer-Kaup-Kupershmidt equation is derived. From the variable separation solution and by selecting appropriate functions, a new class of (2+1)-dimensional nonpropagating solitary waves are found. The novel features exhibited by these new structures are first revealed.     

Algebraic-Geometric Solution to （2＋1）-Dimensional Sawada-Kotera Equation

Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve. Explicit algebraic-geometric solution is obtained on the basis of a deeper understanding of the KdV hierarchy.     

(1+2)-Dimensional model of the early universe

An anisotropic cosmological model is obtained by solving (1+3)-dimensional field equations. The topology of the model isR ¹ M ² S ¹, whereR ¹ is the real line (time axis),M ² is 2-dimensional space, andS ¹ is the circle. Employing the method of Kaluza-Klein type compactification onS ¹ and one-loop quantum correction to scalar fields, an effective (1+2)-dimensional gravity is obtained. The resulting (1+2)-dimensional cosmological model of the early universe is derived.     

Symmetries of a （2＋1）-Dimensional Toda-like Lattice

Lie group technique for solving differential-difference equations is applied to a new (2 1)-dimensional Toda-like lattice. An infinite dimensional Lie algebra and the corresponding commutation relations are obtained.     

Symmetries of a (2+1)-Dimensional Toda-like Lattice

Lie group technique for solving differential-difference equations is applied to a new (2 1)-dimensional Toda-like lattice. An infinite dimensional Lie algebra and the corresponding commutation relations are obtained.     

Variable Separation Solutions in (1+1)-Dimensional and (3+1)-Dimensional Systems via Entangled Mapping Approach

In this paper, the entangled mapping approach (EMA) is applied to obtain variable separation solutions of (1 1)-dimensional and (3 1)-dimensional systems. By analysis, we firstly find that there also exists a common formula to describe suitable physical fields or potentials for these (1 1)-dimensional models such as coupled integrable dispersionless (CID) and shallow water wave equations. Moreover, we find that the variable separation solution of the (3 1)-dimensional Burgers system satisfies the completely same form as the universal quantity U1 in (2 1 )-dimensional systems. The only difference is that the function q is a solution of a constraint equation and p is an arbitrary function of three independent variables.