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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.

With the help of the symbolic computation system Maple and an expanded projective Riccati equation approach, we obtain some new rational explicit solutions with three arbitrary functions for the (2+1)-dimensional Boiti-Leon-Pempinelli system, including Weierstrass function solutions, solitary wave solutions and trigonometric function solutions. From these, several y-periodic soliton localized excitations are constructed and some evolution properties of these novel y-periodic localized structures are discussed.

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.

Variable-coefficient Sawada–Kotere equation is researched. By the means of modified mapping method, we establish a mapping relation between the known solutions of elliptic functional equation and the unknown solutions of variable-coefficient Sawada–Kotere equation. Based on the relation, we easily deduce abundant exact solutions of Jacobi elliptic function and of hyperbolic function to variable-coefficient Sawada–Kotere equation. The merit of our method is that, without much extra effort, we circumvent integration and directly get the above all solutions in an uniform way.

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.

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.