Formula omitted]-symmetric nonlocal Davey–Stewartson I equation: Soliton solutions with nonzero background

Abstract

Various solutions to the PT-symmetric nonlocal Davey–Stewartson (DS) I equation with nonzero boundary condition are derived by constraining different tau functions of the Kadomtsev–Petviashvili hierarchy combined with the Hirota bilinear method. From the first type of tau functions of the nonlocal DS I equation, we construct: (a) general line soliton solutions sitting on either a constant background or on a background of periodic line waves and (b) general lump-type soliton solutions. We find two generic types of line solitons that we call usual line solitons and new-found ones. The usual line solitons exhibit elastic collisions, whereas the new-found ones, in the evolution process, change their waveforms from an antidark (dark) shape to a dark (antidark) one. The general lump-type soliton solutions describe the interaction between 2N-line solitons and 2N-lumps, which give rise to different dynamical scenarios: (i) fusion of line solitons and lumps into line solitons, (ii) fission of line solitons into lumps and line solitons, and (iii) a combination of fusion and fission processes. By constraining another type of tau functions combined with the long wave limit method, periodic line waves, rogue waves, and semi-rational solutions to the nonlocal DS I equation are obtained in terms of determinants whose matrix elements have simple algebraic expressions. Finally, different types of general solutions of the nonlocal nonlinear Schrödinger equation, namely general higher-order breathers and mixed solutions consisting of higher-order breathers and rogue waves sitting on either a constant background or on a background of periodic line waves are obtained as reductions of the corresponding solutions of the nonlocal DS I equation.