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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.

With the aid of computerized symbolic computation, the new modified Jacobi elliptic function expansion method for constructing exact periodic solutions of nonlinear mathematical physics equation is presented by a new general ansatz. The proposed method is more powerful than most of the existing methods. By use of the method, we not only can successfully recover the previously known formal solutions but also can construct new and more general formal solutions for some nonlinear evolution equations. We choose the (3+1)-dimensional Kadomtsev–Petviashvili equation to illustrate our method. As a result, twenty families of periodic solutions are obtained. Of course, more solitary wave solutions, shock wave solutions or triangular function formal solutions can be obtained at their limit condition.

By using the equivalent particle theory, the adiabatic approximation solutions of the Korteweg–de Vries type equation (including KdV equation, cylindrical KdV equation and spherical KdV equation) in dust ion-acoustic solitary waves were obtained. The method can be extended to other nonlinear evolution equations.

By using reductive perturbation method, super KdV equations are changed into ordinary KdV equations, small amplitude perturbation solutions are obtained.

One soliton of particle velocity and its amplitude (maximum particle velocity of one soliton) in Toda lattice is given analytically. It has also been known numerically that the maximum particle velocity (when the collision of two solitons reaches their maximum, we define Vn at this time as its maximum particle velocity) during the collision of two solitons moving in the same direction is equal to the difference between the amplitudes of two solitons if the difference is large enough; however, the maximum particle velocity is equal to the adding up of the amplitudes of two solitons moving in the opposite directions. The relationship between the maximum value of e-rn-1 and their initial amplitude of e-rn-1 is also given analytically in Toda lattice if the amplitudes of the two solitons are the same and their moving directions are opposite. Compared with the Boussinesq equation, there are differences between the Toda lattice equation and the Boussinesq equation for solitons during the collision.

By using the reductive perturbation method, the carrier wave modulation propagating in a weakly viscous fluid-filled elastic tube is investigated. It is shown that such a process can be described by equation Aτ + iBAξξ + iC|A|2 A + DA = 0 and the amplitude of envelope soliton will decrease as it propagates away.

Considering the variation of dust charges, we have analytically studied the governing equation for the system with the same model as that for cold dust acoustic waves in a demagnetized plasma but with the contribution of hot dust. The result indicates that the governing equation is also a Kortweg-de Vries equation, although its amplitude and width will be smaller compared with the cold dust case.

With the aid of computer algebraic system “Mathematica” and by using the hyperbola function method,the exact and solitary wave solutions of the combined KdV equation are obtained,including the bell-like and the kink solitary wave solutions.The thought based on the hyperbola function method is extended,and more explicit and exact solutions are successfully derived,including shock wave type solutions with break point and exponential function type solutions.This method can be used also to solve other nonlinear evolution equation(s).