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A new approach is presented by means of a new general ansatz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, discretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.

The generalized transformation method is utilized to solve three-dimensional Nizhnik–Novikov–Veselov equation and construct a series of new exact solutions including kink-shaped and bell-shaped soliton solutions, trigonometric function solutions, and Jacobi elliptic doubly periodic solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh methods and Jacobi function method, the method we used here gives more general exact solutions without much extra effort.

By using the extended hyperbolic function method, we have studied a quintic discrete nonlinear Schr?dinger 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.

More new exact solutions for a class of nonlinear coupled differential equations are obtained by using a direct and efficient hyperbola function transform method based on the idea of the extended homogeneous balance method.

By the application of the extended homogeneous balance method, we derive an auto-Backlund transformation (BT) for (2+1)-dimensional variable coefficient generalized KP equations. Based on the BT, in which there are two homogeneity equations to be solved, we obtain some exact solutions containing single solitary waves.

Using a further modified extended tanh-function method, rich new families of the exact solutions for the (2+1)-dimensional Broer–Kaup (BK) system, comprising the non-traveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained.

Because it is hard to calculate the TOPMD statistical properties by analytic method, in this paper the modulator method is adopted. So based on polarization-maintaining fiber model, the vector's property and its components are studied by using the Jones Transmission Matrix method. The study show that a 1000-pieces PMF model can represent a purpose very well. Even if we add more pieces in the study model, the histogram of properties have small changes. We can take the simulation result as the theory result. A nPC+nPMF model is used to build the third-order PMDE. The statistical properties of the PMDE is in line with the theory results. This study provides the basis for the TOPMD adaptive compensation.

In order to reduce the number of control instructions DSP issued and the consuming time of PMD compensation, an algorithm processing and logic control unit for first-order and second-order PMD compensation is designed. It plays the functions of DSP's algorithm processing and FPGA's logic control adequately. The configuration of the unit, detailed hardware design and experiment result are also given in the paper. The average consuming time of the unit is even shorter than that of existing DSP compensation module. And it has a better compensation effect.