Semi-rational solutions for the (2+1)-dimensional nonlocal Fokas system

The (2+1)-dimensional [(2+1)d] Fokas system is a natural and simple extension of the nonlinear Schrödinger equation (see Eq. (2) in Fokas, 1994). In this letter, we introduce its PT-symmetric version, which is called the (2+1)d nonlocal Fokas system. The N-soliton solutions for this system are obtained by using the Hirota bilinear method whereas the semi-rational solutions are generated by taking the long-wave limit of a part of exponential functions in the general expression of the N-soliton solution. Three kinds of semi-rational solutions, namely (1) a hybrid of rogue waves and periodic line waves, (2) a hybrid of lump and breather solutions, and (3) a hybrid of lump, breather, and periodic line waves are put forward and their rather complicated dynamics is revealed.