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In this study the general variable separated approach is successfully extended to the (2 + 1)-dimensional physical model. An universal formula involving arbitrary number of variable separated functions is obtained. Because of the existence of the arbitrary functions in the universal formula, new exact quasi-periodic and non-periodic solutions for the (2 + 1)-dimensional nonlinear systems are demonstrated both analytically and graphically by means of the Jacobi elliptic functions with the space–time-dependent modulus. Some novel features or interesting behaviors are revealed.

In this paper, we present a new approach for constructing exact solutions to nonlinear differential-difference equations (NLDDEs). By applying the new method, we have studied the saturable discrete nonlinear Schrodinger equation (SDNLSE) and obtained a number of new exact localized solutions, including 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, provided that a special relation is bound on the coefficients of the equation among the solutions obtained.

By means of a special variable separation approach, a general solution with some arbitrary functions of the three-dimensional Nizhnik–Novikov–Veselov equation is derived. Based on this solution, abundant non-propagating solitons (sometimes solitary waves may be more accurate), such as dromion, ring, peakon, compacton, and foldon, etc. are found by selecting appropriate functions.

In this paper, a time-delayed feedback control method is applied to congestion control in order to eliminate oscillations in the Internet. The stability of the proposed control method is demonstrated based on frequency-domain analysis. The effectiveness of the method is illustrated using simulation.

In this paper, based on a new intermediate transformation, a more general variable coefficient algebraic method is proposed. The efficiency of the method is demonstrated on the Broer–Kaup–Kupershmidt equations. As a result, several new families of exact solutions of physical interest are obtained. The method can be applied to other nonlinear evolution equations in mathematical physics.

The variable separated approach is successfully extended to the (2 + 1)-dimensional modified Nizhnik–Novikov–Vesselov system. Abundant localized coherent solutions are obtained. These exact solutions describe the interactions of different types of soliton. Detail behaviors of these interactions are illustrated both in analytically and in graphically.

By applying a special Backlund transformation, a general variable separation solution of the (2 + 1)-dimensional nonlinear Schrodinger equation is derived. Based on the quite universal variable separation solution and by selecting appropriate functions, a new types of interaction structure between the multi-valued and the single-valued solitary waves, such as fractal semifolded localized structure is investigated in detail and find the interactions possess some novel and interesting features.

A generalized variable-coefficient algebraic method is proposed to construct several new families of exact solutions of physical interest for the (3 + 1)-dimensional Kadomtsev–Petviashvilli (KP) equation with variable coefficients. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with most existing tanh method, the extended tanh method, the Jacobi elliptic function method or the algebraic method, the proposed method gives new and more general solutions.

In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like, period-form solutions of nonlinear evolution equations (NEEs). Compared with most of the existing tanh-function method, extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By using this method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. Make use of the method, we study the (3 + 1)-dimensional potential-YTSF equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions’ soliton-like solutions, singular soliton-like solutions, periodic form solutions.

Starting from a quite universal formula, which is valid for some quite universal (2 + 1)-dimensional physical models, the interactions among different types of solitary waves like peakons, dromions, and compactons are investigated both analytically and graphically. Some novel features or interesting behaviors are revealed. The results show that the interactions for peakon–antidromion, compacton–antidromion, and antipeakon–compacton may be completely inelastic or not completely elastic.

In high dimensions there are abundant coherent soliton excitations. From the variable separation solutions for the (2 + 1)-dimensional Broer–Kaup system, three kinds of new localized excitations in this system are obtained. Some interesting novel features of these structures are revealed.