Non-compatible partially PT symmetric Davey–Stewartson system: Rational and semi-rational solution with nonzero background

The Davey–Stewartson (DS) system is an integrable extension of Schrödinger equation derived from 2 × 2 compatible linear pair, which describes the evolution of optical wave packet in nonlinear optics. This paper introduces its PT symmetric version, namely, the non-compatible partially PT symmetric DS system. “Non-compatible” means the first two equations of the system are inconsistent that the potential field does not need to satisfy certain additional constraints like the compatible case, which renders more freedom of constructing soliton solution. In addition, breather, a hybrid of breather and periodic wave are constructed via Hirota’s method. At the same time, the rational and semi-rational solutions are obtained by applying the “long wave” limit to the general form of N-soliton solution. The rational solution can be classified as lump wave, and two-extremum line rogue wave that appears either bright or dark. The semi-rational solution can be classified as a hybrid of lump wave and periodic wave, a hybrid of lump wave and Akhmediev breather, a hybrid of lump wave, Akhmediev breather, and periodic wave. The proposed technique may also help solve other nonlinear systems, and it is worth studying other symmetric reductions.