Some new exact solutions of Jacobian elliptic function about the generalized Boussinesq equation and Boussinesq-Burgers equation

By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.     

Some New Exact Solutions of Jacobian Elliptic Function of Petviashvili Equation

By using the modified mapping method, we find new exact solutions of the Petviashvili equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.     

Periodic Solutions for a Class of Nonlinear Differential-Difference Equations

In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for three nonlinear differential-difference equations are obtained.     

利用二阶雅可比行列式确定二维重构系统的延迟时间

提出了二阶雅可比行列式的第一极值来确定二维重构系统的延迟时间，它比用互信息函数第一极小确定延迟时间等方法，可以给出更多信息，文中以强迫Ｂｒｕｓｓｅｌａｔｏｒ方程系统的为例进行了讨论。     

Poincaré Map Based on Splitting Methods

Firstly, by using the Liouville formula, we prove that the Jacobian matrix determinants of splitting methods are equal to that of the exact flow. However, for the explicit Runge-Kutta methods, there is an error term of order p ＋ I for the Jacobian matrix determinants. Then, the volume evolution law of a given region in phase space is discussed for different algorithms. It is proved that splitting methods can exactly preserve the sum of Lyapunov exponents invariable. Finally, a Poincaré map and its energy distribution of the Duffing equation are computed by using the second-order splitting method and the Heun method （a second-order Runge-Kutta method）. Computation illustrates that the results by splitting methods can properly represent systems＇ chaotic phenomena.     

Lame Function and Its Application to Some Nonlinear Evolution Equations

In this paper, based on the Lame function and Jacobi emptic function, the perturbation method is appliedto some nonlinear evolution equations to derive their multi-order solutions.     

常用数学工具在热力学关系式证明中的应用

对热力学中常用的数学工具如多元函数的导数及其全微分、雅可比行列式、勒让德变换、复合函数求导等进行了简单介绍,并由热力学基本规律出发,在充分分析基本热力学关系式特点的基础上,较系统地归纳出6种解决热力学关系式证明的方法,对每一种方法都给出了适用的要点,并以典型例题进行了说明.     

动力系统时间延迟重构变换的行为

利用延迟重构变换的雅可比行列式,分析了动力系统重构中的拓朴性质。讨论了延迟时间的选择和重构变量本身的条件稳定性对重构拓朴的影响。利用条件Lyapunov指数分析了重构变量本身的好坏。文中以强迫Brusselator系统为例进行了讨论     

General Jacobian elliptic function expansion method and its applications

An extended Jacobian elliptic function expansion method is presented and successfully applied to the nonlinear Schr?dinger (NLS) equation and Zakharov equation. We obtain some new solutions besides Fu et al's results. The results show that our method is more powerful to construct Jacobian elliptic function and can be applied to other nonlinear physics systems.     

Periodic Solutions for Two Coupled Nonlinear-Partial Differential Equations

In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for two coupled nonlinear partial differential equations are obtained.     

New Doubly Periodic Solutions for the Coupled Nonlinear Klein-Gordon Equations

By using the general solutions of a new coupled Riccati equations, a direct algebraic method is described to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear Klein-Gordon equations.It is shown that more doubly periodic solutions and the corresponding solitary wave solutions and trigonometric function solutions can be obtained in a unified way by this method.     

New Weierstrass Semi-rational Expansion Method to Doubly Periodic Solutions of Soliton Equations

Based on the Weierstrass elliptic function equation, a new Weierstrass semi-rational expansion method and its algorithm are presented. The main idea of the method changes the problem solving soliton equations into another one solving the corresponding set of nonlinear algebraic equations. With the aid of Maple, we choose the modified KdV equation, (2+1)-dimensional KP equation, and (3+1)-dimensional Jimbo-Miwa equation to illustrate our algorithm. As a consequence, many types of new doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Moreover the corresponding new Jacobi elliptic function solutions and solitary wave solutions are also presented as simple limits of doubly periodic solutions.     

New Periodic Wave Solutions to Generalized Klein-Gordon and Benjamin Equations

By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for generalized Klein-Cordon equation and Benjamin equation, which cannot be found in previous work. This method also can be used to find new periodic wave solutions of other nonlinear evolution equations.     

Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to(1+1)-Dimensional Dispersive Long Wave Equation

In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and O, the hyperbolic function solutions （including the solitary wave solutions） and trigonometric solutions are also given respectively.     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and 0, the hyperbolic function solutions (including the solitary wave solutions) and trigonometric solutions are also given respectively.     

Extended F-Expansion Method and Periodic Wave Solutionsfor Klein-Gordon-Schrödinger Equations

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrödinger equations are obtained. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.     

Periodic Wave Solutions to Dispersive Long-Wave Equations in (2+1)-Dimensional Space

Periodic wave solutions to the dispersive long-wave equations are obtained by using the F-expansion method, which can be thought of as a generalization of the Jacobi elliptic function method. In the limit case, solitary wave solutions are obtained as well.     

Wave Localization of Doubly Periodic Guided-mode Resonant Grating Filters

It is known that a doubly periodic guided-mode resonant grating (GMRG) filter has a broad angular selectivity with a narrow spectral bandwidth. This means that the doubly periodic GMRG filter operates for small beam diameter and grating area. This report describes the wave localization in the doubly periodic GMRG filter. We investigated the spread area of light waves in the waveguide layer and the accumulation of field energy by numerical simulation using the finite differential time domain (FDTD) method. Simulation results showed that, in the case of a doubly periodic GMRG filter with a Q factor of 600, the field energy is spread over an area 5 um in width, which corresponds to the expected value from the angular tolerance. And the magnitude of the field energy in the waveguide layer was Q factor times greater than the incident energy. On the other hand, a singly periodic GMRG filter with the same Q factor spread the field energy over an area 72 urn in width. This filter does not work for a small size structure or a small diameter light beam.     

Periodic Wave Solutions to Dispersive Long-Wave Equations in （2＋1）-Dimensional Space

Periodic wave solutions to the dispersive long-wave equations are obtained by using the F-expansion method, which can be thought of as a generalization of the Jacobi elliptic function method. In the limit case, solitary wave solutions are obtained as well.