Applications of Jacobi Elliptic Function Expansion Method for Nonlinear Differential-Difference Equations

The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the application of the Jacobi elliptic function expansion method. As a result, three types of periodic wave solutions including Jacobi elliptic sine function, Jacobi elliptic cosine function and the third elliptic function solutions are obtained. It is shown that the shock wave solutions and solitary wave solutions can be obtained at their limit condition.

Solving Nonlinear Wave Equations by Elliptic Equation

The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.

Doubly Periodic Wave Solutions of Jaulent-Miodek Equations Using Variational Iteration Method Combined with Jacobian-function Method

One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.

New Exact Solutions for （2＋1）-Dimensional Breaking Soliton Equation

New exact solutions in terms of the Jacobi elliptic functions are obtained to the (2+1)-dimensional breakingsoliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solutionsand triangular periodic wave solutions are obtained.

Equatorial Rossby Solitary Wave Under the External Forcing

A simple shallow-water model with influence of external forcing on a β-planeis applied to investigate the nonlinear equatorial Rossby waves in a shear flow. By theperturbation method, the extended variable-coefficient KdV equation under an external forcing isderived for large amplitude equatorial Rossby wave in a shear How. And then various periodic-likestructures for these equatorial Rossby waves are obtained with the help of Jacobi ellipticfunctions. It is shown that the external forcing plays an important role in various periodic-likestructures.

New Periodic Wave Solutions, Localized Excitations and Their Interaction Properties for （3＋1）-Dimensional Jimbo-Miwa Equation

Based on the multi-linear variable separation approach, a class of exact, doubly periodic wave solutions for the （3＋1）-dimensional Jimbo-Miwa equation is analytically obtained by choosing the Jacobi elliptic functions and their combinations. Limit cases are considered and some new solitary structures （new dromions） are derived. The interaction properties of periodic waves are numerically studied and found to be inelastic. Under long wave limit, two sets of new solution structures （dromions） are given. The interaction properties of these solutions reveal that some of them are completely elastic and some are inelastic.

Breather Lattice Solutions to Negative mKdV Equation

In this paper,dependent and independent variable transformations are introduced to solve the negativemKdV equation systematically by using the knowledge of elliptic equation and Jacobian elliptic functions.It is shownthat different kinds of solutions can be obtained to the negative mKdV equation,including breather lattice solution andperiodic wave solution.

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.

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 rational form solutions to mKdV equation

In this paper, new basic functions, which are composed of three basic Jacobi elliptic functions, are chosen as components of finite expansion. This finite expansion can be taken as an ansatz and applied to solve nonlinear wave equations. As an example, mKdV equation is solved, and more new rational form solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form, and so on.

Exact Periodic-Wave Solutions to Nizhnik-Novikov-Veselov Equation

Exact periodic-wave solutions to the generalized Nizhnik-Novikov-Veselov (NNV) equation are obtained by using the extended Jacobi elliptic-function method, and in the limit case, the solitary wave solution to NNV equation are also 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.

Aging gracefully: social engagement joins exercise and enrichment as a key lifestyle factor in resistance to age-related cognitive decline

Cognitive impairment is a consequence of the normal aging process that effects many species, including humans and rodent models. Decline in hippocampal memory function is especially prominent with age and often reduces quality of life. As the aging population expands, the need for interventional strategies to prevent cognitive decline has become more pressing. Fortunately, several major lifestyle factors have proven effective at combating hippocampal aging, the most well-known of which are environmental enrichment and exercise. While the evidence supporting the beneficial nature of these factors is substantial, a less well-understood factor may also contribute to healthy cognitive aging: social engagement. We review the evidence supporting the role of social engagement in preserving hippocampal function in old age. In elderly humans, high levels of social engagement correlate with better hippocampal function, yet there is a dearth of work to indicate a causative role. Existing rodent literature is also limited but has begun to provide causative evidence and establish candidate mechanisms. Summed together, while many unanswered questions remain, it is clear that social engagement is a viable lifestyle factor for preserving cognitive function in old age. Social integration across the lifespan warrants more investigation and more appreciation when designing living circumstances for the elderly.

Elliptic Equation and Its Direct Applications to Nonlinear Wave Equations

Elliptic equation is taken as an ansatz and applied to solve nonlinear wave equations directly. More kinds of solutions are directly obtained, such as rational solutions, solitary wave solutions, periodic wave solutions and so on.It is shown that this method is more powerful in giving more kinds of solutions, so it can be taken as a generalized method.

Recent Changes Occurred in the Terminus of the Debriscovered Bilafond Glacier in the Karakoram Himalayas Using Remotely Sensed Images and Digital Elevation Models(1978-2011)

Recent changes occurred in terminus of the debris-covered Bilafond Glacier in the Karakoram Range in the Himalayas, Northern Pakistan was investigated in this research. Landsat MSS, TM and ETM+ images were used for this study. Digital elevation models derived from ASTER GDEM and SRTM were also utilized. Visible, infrared and thermal infrared channels were utilized in order to get accurate glacier change maps. Three methods were tried to map this debris-covered glacier in this research. The glacier has been mapped successfully and the changes in the glacier terminus from 1978 to 2011 have been calculated. Manual, semi-automatic and thermal methods were found to give similar results. It was found that the glacier has undergone serious ablation during this period despite of the fact that many of the larger glaciers in the Hindu Kush and Karakoram mountain regions in the Upper Indus Basin were reported to be expanding. The terminus has been moved back about 600 meters during this period and there was an abrupt change in the glacier terminus during 1990-2002. We propose that debris thickness is not the only factor that influences the glacier ablation but the altitude of the debris-covered glacier as well. Many glaciers in the Karakoram region reported to be expanding were having higher altitudes compared to the study area.

Internal Variable Formulations of Static and Dynamic Elastic-Plastic Problems Using Liapunov Functions： Algorithmic Aspects

The fundamental problem of an elastic-plastic body subjected to incremental loading is reviewed using a compact internal variable approach based on work carried out at the University of Cape Town in which a quadratic functional was developed for the free energy using Taylor series. Now the departure from that approach is the focus on developing the Liapunov function for the nonlinear differential equations of motion. Static and dynamic equations of motion are derived and shown to meet the requirements of the Liapunov function. As a consequence, time integration parameters that are used in the discrete formulations are easily obtained based on the same requirements. The resulting generalized Newton-Raphson scheme is stable in the sense of Liapunov＇s direct method.

Effect of Lead Toxicity on Germination and Enzymatic Activity of Durum Wheat Triticum durum Desf.

The aim of this work is to identify the effect of lead on germinal parameters and the antioxidant enzyme activities （lipase, peroxidase and catalase） in durum wheat Triticum durum Desf. cv （waha, vitron and gta） exposed to the concentrations of 0, 0.15, 0.25 and 0.3 g/L of Pb （NO3）2 during germination process. The obtained results showed that lead reduced the germination, root and aerial biomass. The concentration of 0.3 g/L inhibited completely the germination of the three varieties. It also slowed lipase activity, the degradation of lipids of the seed＇s reserves and disrupted the metabolism of peroxidase and catalase. Concerning the behavior of the three varieties studied, it appears that the Vitron is the best predisposed variety to stand against lead stress by its strong antioxidant defense system.

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.

Entropy Production along Dominant Pathway of Nonequilibrium Phase Transition in Mesoscopic Chemical System

We consider a bistable mesoscopic chemical reaction system and calculate entropy produc- tion along the dominant pathway during nonequilibrium phase transition. Using probability generating function method and eikonal approximation, we first convert the chemical master equation into the classical Hamilton-Jacobi equation, and then find the dominant pathways between two steady states in the phase space by calculating zero-energy trajectories. We find that entropy productions are related to the actions of the forward and backward dominant pathways. At the coexistence point where the stabilities of the two steady states are equiv alent, both the system entropy change and the medium entropy change are zero; whereas at non-coexistence point both of them are nonzero.