Time-like boundary conditions in the NLS model

We focus on the non-linear Schrödinger model and we extend the notion of space-time dualities in the presence of integrable time-like boundary conditions. We identify the associated time-like “conserved” quantities and Lax pairs as well as the corresponding boundary conditions. In particular, we derive the generating function of the space components of the Lax pairs in the case of time-like boundaries defined by solutions of the reflection equation. Analytical conditions on the boundary Lax pair lead to the time like-boundary conditions. The time-like dressing is also performed for the first time, as an effective means to produce the space components of the Lax pair of the associated hierarchy. This is particularly relevant in the absence of a classical r-matrix, or when considering complicated underlying algebraic structures. The associated time Riccati equations and hence the time-like conserved quantities are also derived. We use as the main paradigm for this purpose the matrix NLS-type hierarchy.