The general integrable inhomogeneous Dirac–Manakov equations: Darboux-dressing transformation and N-soliton solutions

By means of the zero-curvature equation and two sets of Lenard recursion sequences, we construct a nonisospectral generalized 3 × 3 Ablowitz–Kaup–Newell–Segur (AKNS) integrable hierarchy. The general inhomogeneous Dirac–Manakov equations with temporally and spatially modulated coefficients are derived. Under some special reductions, the two-component nonlinear Schrödinger (NLS) or Manakov equations with helicoidal spin-orbit (SO) coupling, Rabi coupling, and mixed helicoidal SO and Rabi couplings are obtained. In particular, we propose several novel inhomogeneous two-component NLS equations which have variable helicoidal SO and Rabi couplings as well as external potentials. The N -fold Darboux-dressing transformation related to the nonisospectral generalized 3 × 3 AKNS matrix eigenvalue problem is established, and further the N -soliton solution represented in a compact determinant form is given. The classification of solitons for a two-component NLS equations with temporally modulated mixed helicoidal SO and Rabi couplings is discussed in detail. Specifically, three types of solitons including beating, bell-shaped and multi-peak solitons as well as their parabolic, periodic, exponential and kinked deformations, and some unconventional nonlinear superpositions are presented.