Rational and semi-rational solutions of the y-nonlocal Davey-Stewartson I equation

The nonlocal Davey-Stewartson (DS) I equation with a parity-time-symmetric potential with respect to the y-direction, which is called the y-nonlocal DS I equation, is a two-dimensional analogue of the nonlocal nonlinear Schrödinger (NLS) equation. The multi-breather solutions for the y-nonlocal DS I equation are derived by using the Hirota bilinear method. Lump-type solutions and hybrid solutions consisting of lumps sitting on periodic line waves are generated by long wave limits of the obtained soliton solutions. Also, various types of analytical solutions for the nonlocal NLS equation with negative nonlinearity, including both the Akhmediev breathers and the Peregrine rogue waves sitting on periodic line waves, can be generated with appropriate constraints on the parameters of the obtained exact solutions of the y-nonlocal DS I equation. Particularly, we show that a family of hybrid solitons describing the Peregrine rogue wave that coexists with the Akhmediev breather, both of them sitting on a spatially-periodic background can be thus obtained.