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Based on active control theory, anti-synchronization between two different hyperchaotic systems is investigated. The sufficient conditions for achieving anti-synchronization of two different hyperchaotic systems are derived. Moreover, numerical simulations are presented for hyperchaotic Lorenz–Chen system, hyperchaotic Lorenz–Lu system, and hyperchaotic Chen–Lu system to verify the effectiveness and feasibility of the proposed anti-synchronization scheme.

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 potential method and the perturbation method under the condition of small amplitude and shallow water waves, we analytically get the -type equation for a viscous shallow water. It indicates that for one soliton-like solution, its amplitude will decrease as it propagates away due to the viscous effects of water.