Considerations on conserved quantities and boundary conditions of the 2+12+1-dimensional nonlinear Schrödinger equation

In this study, we consider a natural integrable generalization of the defocusing cubic nonlinear Schrödinger equation to two dimensions and we classify the admissible boundary conditions. In particular, we determine whether the classical physical observables are conserved: mass, momentum, and Hamiltonian. We find that this is the case when a certain integral (the mass constraint) vanishes. The vanishing of the mass constraint, and thus the existence of conserved quantities, is contingent on the boundary conditions adopted. In particular, under decaying boundary conditions, the Hamiltonian is not necessarily conserved.