Varieties of exact solutions for the (2+1)- dimensional nonlinear schrödinger equation with the trapping potential

The separation of variables and Hirota's bilinearization technique are applied to solve, in Cartesian coordinates, the (2+1)-dimensional nonlinear Schrödinger (NLS) equation with specially chosen external trapping potentials. The inverse problem is introduced and solved, aimed at finding the trapping potential that supports the prescribed form of the solution. It consists in appropriately selecting two arbitrary functions, which describe the exact solutions of the NLS equation, and defining the trapping potential through Hirota's bilinear forms, which involve the two functions. A variety of localized wave structures, such as single and multiple dromions ("dromion lattices"), ring, parabolic, and breather modes, and two-dimensional square-shaped peakons, are found. The stability of these solutions is checked by propagating them numerically for long distances.