Ultracold atoms can be trapped in strongly confined low dimensional trap geometries like (1D) cigars and (2D) pancakes: this allows for to experimentally simulate Hamiltonians having strongly correlated ground states.  The 1D Bose gas is well described by the so called Lieb-Liniger Hamiltonian: in the talk I will discuss a recently proposed method to compute expectation values and other physical quantities.  The method is based on the observation that the S-matrix and the Lagrangian of the Lieb-Liniger model can be obtained as a non-relativistic limit of an integrable relativistic field theory, the sinh-Gordon model.  In the final part of the talk, I will briefly discuss the effect of an experimentally realizable artificial non-abelian magnetic field acting on a 2D ultracold atomic gas, showing that it is possible to tune the Hamiltonian parameters in order to have non-abelian excitations.