With the help of the symbolic computation system Maple and an improved mapping method and a variable separation method, a series of new exact solutions (including solitary wave solutions and periodic wave solutions) to the (2+1)-dimensional general Nizhnik-Novikov-Veselov (GNNV) system is derived. Based on the derived solitary wave solution, we obtain some chaotic patterns.

With the help of the symbolic computation system Maple and the mapping approach and a linear variable separation approach, a new family of exact solutions of the (1+1)-dimensional Burgers system is derived. Based on the derived solitary wave solution, some novel bell wave and kind wave excitations are investigated.

The mapping approach is a powerful tool to looking for the exact solutions for nonlinear partial differential equations. In this paper, using an improved mapping approach, a series of exact solutions (including solitary wave solutions and periodic wave solutions) of the (2+1)-dimensional dissipative Zabolotskaya Khokhlov (DZK) system is derived. Based on the derived solitary wave solution, we obtain some folded localized excitations of the DZK system.

With the mapping method and a variable separation method, a series of variable separation solutions to the extended (2+1)-dimensional shallow water wave (ESWW) system is derived. Based on the derived solitary wave excitation, some chaotic behaviors are investigated.

The dynamics of a two-dimensional vortex system in superconductors with periodic artificial columnar pinning is studied. The ground state at field B = 3Bf can be either anisotropic or isotropic, dependent on pinning strength and size, here Bf is the matching field where the number of vortices equals that of pins. The transport curves are dependent on the ground vortex structures and anisotropic ground structure may result in anisotropic velocity-force curve. Results indicate that the ground structure can be detected from the transport property. We also discover that a jump in velocity-force curve accompanies a structure transition.