  In this paper we study a suitable limit of integrable QFT with the aim to
identify continuous non-relativistic integrable models with local interactions.
This limit amounts to sending to infinity the speed of light c but
simultaneously adjusting the coupling constant g of the quantum field theories
in such a way to keep finite the energies of the various excitations. The QFT
considered here are Toda Field Theories and the O(N) non-linear sigma model. In
both cases the resulting non-relativistic integrable models consist only of
Lieb-Liniger models, which are fully decoupled for the Toda theories while
symmetrically coupled for the O(N) model. These examples provide explicit
evidence of the universality and ubiquity of the Lieb-Liniger models and, at
the same time, suggest that these models may exhaust the list of possible
non-relativistic integrable theories of bosonic particles with local
interactions.
