Spatiotemporal superposed rogue-wave-like breathers in a (3 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation

A one-to-one relation between a variable-coefficient (3+1)-dimensional nonlinear Schrodinger equation with linear and parabolic potentials and the standard nonlinear Schrodinger equation is presented, and then superposed rogue-wave-like breather solution is obtained. These explicit expressions, describing the evolution of the amplitude, width, center and phase, imply that the diffraction, nonlinearity and gain/loss parameters interplay together to influence evolutional characteristics above. Moreover, the controllable mechanism for fast excitation, maintenance, restraint and recurrence of breather is studied. We also provide an experimental scheme to observe these phenomena in future experiments.