An extended functional transformation method and its application in some evolution equations

In this paper, an extended functional transformation is given to solve some nonlinear evolution equations. This function, in fact, is a solution of the famous KdV equation, so this transformation gives a transformation between KdV equation and other soliton equations. Then many new exact solutions can be given by virtue of the solutions of KdV equation.

(G'/G)-Expansion Method Equivalent to Extended Tanh Function Method

In a recent article [Physics Letters A 372 （2008） 417], Wang et al. proposed a method, which is called the （G′/G）-expansion method, to look for travelling wave solutions of nonlinear evolution equations. The travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations, and the Hirota-Satsuma equations are obtained by using this method. They think the （G′/G）-expansion method is a new method and more travelling wave solutions of many nonlinear evolution equations can be obtained. In this paper, we will show that the （G′/G）-expansion method is equivalent to the extended tanh function method.

Applications of Extended Hyperbolic Function Method for Quintic Discrete Nonlinear SchrSdinger Equation

By using the extended hyperbolic function method, we have studied a quintic discrete nonlinear Schrodinger equation and obtained new exact localized solutions, including the discrete bright soliton solution, dark soliton solution, bright and dark soliton solution, alternating phase bright soliton solution, alternating phase dark soliton solution, and alternating phase bright and dark soliton solution, if a special relation is bound on the coefficients of the equation.

Jacobian Elliptic Function Method and Solitary Wave Solutions for Hybrid Lattice Equation

In this paper, we have successfully extended the Jacobian elliptic function expansion approach to nonlinear differential-difference equations. The Hybrid lattice equation is chosen to illustrate this approach. As a consequence, twelve families of Jacobian elliptic function solutions with different parameters of the Hybrid lattice equation are obtained. When the modulus m → 1 or O, doubly-periodic solutions degenerate to solitonic solutions and trigonometric function solutions, respectively.

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the （2＋1）-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.

Some Integral Inequalities of Simpson Type for Strongly Extended s-Convex Functions

The main purpose of this survey paper is to point out some very recent developments on Simpson’s inequality for strongly extended s-convex function. Firstly, the concept of strongly extended s-convex function is introduced. Next a new identity is also established. Finally, by this identity and H?lder’s inequality, some new Simpson type for the product of strongly extended s-convex function are obtained.

Symbolic Computation of Extended Jacobian Elliptic Function Algorithm for Nonlinear Differential-Different Equations

The Jacobian elliptic function expansion method for nonlinear differential-different equations and its algorithm are presented by using some relations among ten Jacobian elliptic functions and successfully construct more new exact doubly-periodic solutions of the integrable discrete nonlinear Schrodinger equation. When the modulous m → 1or 0, doubly-periodic solutions degenerate to solitonic solutions including bright soliton, dark soliton, new solitons as well as trigonometric function solutions.

New Exact Periodic Solitary-Wave Solutions for New (2+1)-Dimensional KdV Equation

The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions （including kinky periodic solitary-wave solutions, periodic soliton solutions, and cross kink-wave solutions） for the new （2＋1）-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.

Dynamics of solitons in Bose-Einstein condensate with time-dependent atomic scattering length

The evolution of solitons in Bose-Einstein condensates （BECs） with time-dependent atomic scattering length in an expulsive parabolic potential is studied. Based on the extended hyperbolic function method, we successfully obtain the bright and dark soliton solutions. In addition, some new soliton solutions in this model are found. The results in this paper include some in the literature （Phys. Rev. Lett. 94（2005）050402 and Chin. Phys. Lett. 22（2005） 1855）.

NEW PERIODIC SOLUTIONS OF ITO'S 5th-ORDER mKdV EQUATION AND ITO'S 7th-ORDER mKdV EQUATION

Based on the modified Jocobi elliptic function expansion method and the modified extended tanh function method,a new algebraic method is presented to obtain mu ltiple travelling wave solutions for nonlinear wave equations.By using the metho d,Ito's 5th order and 7th order mKdV equations are studied in detail and more new exact Jocobi elliptic function periodic solutions are found.With modulus m→1 or m→0,these solutions degenerate into corresponding solitary wave s olutions,shock wave solutions and trigonometric function solutions.

Discrete doubly periodic and solitary wave solutions for the semi-discrete coupled mKdV equations

In this paper, the improved Jacobian elliptic function expansion approach is extended and applied to constructing discrete solutions of the semi-discrete coupled modified Korteweg de Vries （mKdV） equations with the aid of the symbolic computation system Maple. Some new discrete Jacobian doubly periodic solutions are obtained. When the modulus m →1, these doubly periodic solutions degenerate into the corresponding solitary wave solutions, including kink-type, bell-type and other types of excitations.

New infinite-dimensional symmetry groups for the stationary axisymmetric Einstein-Maxwell equations with multiple Abelian gauge fields

The so-called extended hyperbolic complex （EHC） function method is used to study further the stationary axisymmetric Einstein Maxwell theory with p Abelian gauge fields （EM-p theory, for short）, Two EHC structural Riemann- Hilbert （RH） transformations are constructed and are then shown to give an infinite-dimensional symmetry group of the EM-p theory. This symmetry group is verified to have the structure of semidirect product of Kac-Moody group SU（p ＋ 1, 1） and Virasoro group. Moreover, the infinitesimal forms of these two RH transformations are calculated and found to give exactly the same infinitesimal transformations as in previous author＇s paper by a different scheme, This demonstrates that the results obtained in the present paper provide some exponentiations of all the infinitesimal symmetry transformations obtained before.

Low-Frequency Stability Analysis of Power Electronic Traction Transformer Based Train-Network System

Power electronic traction transformers(PETTs)will be increasingly applied to locomotives in the future for their small volume and light weight.However,similar to conventional trains,PETTs behave as constant power loads and may cause low-frequency oscillation(LFO)to the train-network system.To solve this issue,a mathematical model of the PETT is firstly proposed and verified based on the extended describing function(EDF)method in this paper.In the proposed model,the LLC converter is simplified to an equivalent circuit consisting of a capacitor and a resistor in parallel.It is further demonstrated that the model can apply to various LLC converters with different topologies and controls.Particularly,when the parameter differences between cells are not obvious,the PETT can be simplified to a single-phase rectifier(i.e.,conventional train)by equivalent transformation.Based on the model of PETT,the system low-frequency stability and influential factors are analyzed by using the generalized Nyquist criterion.Lastly,the correctness and accuracy of theoretical analyses are validated by off-line and hardware-in-the-loop simulation results.

Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.

EFFECT OF ARCTIC ICE COVER ON THE MONTHLY TEMPERATURE AND PRECIPITATION IN CHINA

Filtering technique, extended empirical orthogonal function (EEOF), spectrum distribution function and correla- tion analysis have been employed to study the relationship between arctic ice cover (AIC) and monthly mean tempera- ture and precipitation in China. The function of power spectrum density shows that not only a semi-annual and an an- nual oscillation but also a quasi-biennial oscillation can be found in AIC area index series, especially in June, September and November. During the period of analysis, it can also be found that there exists a good correlation between the El Nino events and the AIC area index. An analysis on the EEOF of AIC and the temperature over China exhibits some significant temporal-spatial patterns and a better time-lag interrelationship between them. The results from the correla- tion analysis indicate that the variation of AIC area has a significant influence on the temperature and precipitation in subsequent months over China. In addition, it experiences a quasi-biennial low-frequency oscillation and displays to certain extent some features of propagation.

Visual polysemy and synonymy:toward near-duplicate image retrieval

that are duplicate or near duplicate to a query image.One of the most popular and practical methods in near-duplicate image retrieval is based on bag-of-words(BoW)model.However,the fundamental deficiency of current BoW method is the gap between visual word and image’s semantic meaning.Similar problem also plagues existing text retrieval.A prevalent method against such issue in text retrieval is to eliminate text synonymy and polysemy and therefore improve the whole performance.Our proposed approach borrows ideas from text retrieval and tries to overcome these deficiencies of BoW model by treating the semantic gap problem as visual synonymy and polysemy issues.We use visual synonymy in a very general sense to describe the fact that there are many different visual words referring to the same visual meaning.By visual polysemy,we refer to the general fact that most visual words have more than one distinct meaning.To eliminate visual synonymy,we present an extended similarity function to implicitly extend query visual words.To eliminate visual polysemy,we use visual pattern and prove that the most efficient way of using visual pattern is merging visual word vector together with visual pattern vector and obtain the similarity score by cosine function.In addition,we observe that there is a high possibility that duplicates visual words occur in an adjacent area.Therefore,we modify traditional Apriori algorithm to mine quantitative pattern that can be defined as patterns containing duplicate items.Experiments prove quantitative patterns improving mean average precision(MAP)significantly.