Nonlinear tunnelling effect of combined Kuznetsov–Ma soliton in (3+1)-dimensional \[{{\varvec{\mathcal {PT}}}}\] -symmetric inhomogeneous nonlinear couplers with gain and loss

A (3+1)-dimensional variable-coefficient coupled nonlinear Schrödinger equation in parity-time symmetric inhomogeneous nonlinear couplers with gain and loss is studied, and the \[{\mathcal {PT}}\]-symmetric and \[{\mathcal {PT}}\]-antisymmetric combined Kuznetsov–Ma soliton solutions are obtained via the Darboux transformation method. Nonlinear tunnelling effect of controllable combined Kuznetsov–Ma solitons such as their restraint, maintenance and postpone are discussed when they pass through the dispersion/diffraction barrier and well.