New solutions for the (2 + 1)-dimensional nonlinear Schrӧdinger equation

•The solutions of the (2 + 1)-dimensional nonlinear Schrӧdinger equation are obtained that the double solitons, breather solitons, periodic waves and rational rogue waves.•The collision between two solitons is an elastic collision, and the periodic wave forms a chaotic-periodic banchart.•Two kinds of rogue wave structures are found: breather solitons and rational rogue waves. New exact solutions to the (1 + 1)-dimensional nonlinear Schrӧdinger equation are obtained by means of the exponential function test method. Further, the solutions of the (2 + 1)-dimensional nonlinear Schrӧdinger equation are also obtained by using appropriate transformation and being based on solutions obtained. Among them several arbitrary parameters are involved to generate a large number of wave structures, they are the double solitons, breather solitons, periodic waves and rational rogue waves, which dynamical behavior are shown by digital images.