Rogue Waves in the (2+1)-Dimensional Nonlinear Schrödinger Equation with a Parity-Time-Symmetric Potential**Supported by the National Natural Science Foundation of China under Grant Nos 11271211, 11275072 and 11435005, the Ningbo Natural Science Foundation under Grant No 2015A610159, the Opening Project of Zhejiang Provincial Top Key Discipline of Physics Sciences in Ningbo University under Grant No xkzw11502, and the K. C. Wong

Abstract

The (2+1)-dimension nonlocal nonlinear Schrödinger (NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equation introduced by Ablowitz et al. [Phys. Rev. Lett. 110 (2013) 064105]. General periodic solutions are derived by the bilinear method. These periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again. By taking long wave limits of the obtained periodic solutions, rogue waves are obtained. It is also shown that these line rogue waves arise from the constant background with a line profile and disappear into the constant background again in the plane.