The mapping approach is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. Its remarkable characteristic is that we can have infinitely different ansatzs and thus end up with the abundance of solutions. By an improved mapping approach and a variable separation method, a series of excitations of the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system is derived. Based on the derived solitary wave excitation, we obtain some special localized structures such as peakon solitons and fractal solitons, then we discuss the phenomenon of “chase and collision”.