\(\overline{\partial }\)-dressing method to PT-symmetric multi-component nonlinear Schrödinger equation

PT-symmetry is physically useful as it can overcome losses while still preserving guidance properties of the system. In this paper, the PT-symmetric multi-component nonlinear Schrödinger (PTSMCNLS) equation is derived using the \(\overline{\partial }\) $\overline{\partial }$ -dressing method in two local \(4 \times 4\) $4\times 4$ matrix \(\overline{\partial }\) $\overline{\partial }$ -problems. The relations between the potential of the PTSMCNLS equation and the solution of the \(\overline{\partial }\) $\overline{\partial }$ -problem are established. The explicit N-soliton solutions of the PTSMCNLS equation are obtained. Moreover, the PT-symmetric three-component nonlinear Schrödinger (PTSTCNLS) equation is constructed by virtue of reduction. As applications, some specific soliton dynamical behaviors are theoretically explored and graphically illustrated.