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One of the advantages of the variational iteration method is the free choice of initial guesses. In this paper we use the basic idea of the Exp-function method to construct a generalized trial function with some unknown parameters. The Klein–Gordon equation is used to illustrate the effectiveness and convenience of the method, some new explicit travelling wave solutions are obtained which include bell-type soliton solution, kink-type soliton solution, and generality solitary wave solutions.

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 Exp-function method to construct a generalized trial function with some unknown parameters. The mKdV equation is used to illustrate effectiveness and convenience of the method, some new explicit exact travelling wave solutions are obtained which include bell-type soliton solution, kink-type soliton solution, and periodic wave solution.

Using the variable coefficient generalized projected Ricatti equation expansion method, we present the explicit solutions of the (2 + 1)-dimensional Broer–Kaup (BK) equation. These solutions include Weierstrass function solutions, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Some of these solutions are found for the first time. Because of the two or three arbitrary functions, rich localized excitations can be found.

In this paper, the cubic–quintic nonlinear Schrodinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.

In high dimensions, there are abundant coherent soliton excitations. From the variable separation approach of the celebrated Kadomtsev–Petviashvili(KP) equation and the coupled Burgers equation, several new multisoliton excitations are derived and some novel features or interesting behaviors of these structures are revealed.

Wick-type stochastic KdV equations are researched. By the means of Hermite transformation and mapping method, many new exact Jacobi function solutions of stochastic KdV equations are derived.

In this manuscript, using the variable-coefficient generalized projected Ricatti equation expansion method, we present explicit solutions of the mKDV equation with variable coefficients. These solutions include solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time.

In this paper, using the variable-coefficient F-expansion method, we present a number of Jacobian elliptic function solutions of nonlinear Schrodinger equations with variable-coefficient. Particular cases of these solutions, where the elliptic function modulus equals 1 and 0, are various localized solutions and trigonometric function solutions, respectively. Each of these solutions exists for a certain relation between the parameters of the equation. Therefore, they are particular cases of the complete set of periodic and localized solutions which may exist for this equation. In fact, they can serve as a seeding set of solutions which could be useful in other optical studies.

Using He's homotopy perturbation method (HPM), the modified KdV equation which has not a small parameter is solved. The approximate Jacobi elliptic function solution is obtained. When the modulus of the Jacobian function tends to unit or zero, the corresponding solitary wave solution and trigonometric function solution are obtained. The results reveal that the HPM is very effective, convenient and quite accurate to systems of nonlinear equations.

A Wick-type stochastic Korteweg–de Vries equation is researched. We establish a relationship between the Korteweg–de Vries equation and the elliptic equation. With the help of Hermit transformation and the elliptic equation mapping method, we obtain a series of exact Jacobi elliptic function solutions to the stochastic Korteweg–de Vries equation in the white noise environment.