‘Approximate’ conservation laws of some nonlinear Schrödinger equation involving a spatially extended system consisting of two coupled elements

In this paper, we construct and analyse the conservation laws of a model of a nonlinear Schrödinger equation involving a spatially extended system consisting of two coupled elements. In the first of two cases, we study the space–time system and, separately, the stationary model arising from a standard transformation. In the second case, we analyse a ‘toy model PT-symmetric dimer’ involving a ‘gain-loss amplitude parameter’ for which only approximate conserved forms exist. In this latter case, Noether's theorem is not applicable as it is not variational and the conserved forms are adapted in an ‘approximate’ and novel way due to the nature of the system.