| Abstract: | The fundamental role played by the quantum repulsive Nonlinear Schrödinger (NLS) equation in the evolution of our understanding of the phenomenon of superconductivity in appropriate metals at very low temperatures is surveyed. The first major work was that in 1947 by N. N. Bogoliubov, who studied the very physical 3-space-dimensions problem and super fluidity; and the survey takes the form of an actual dedication to that outstanding scientist who died four years ago. The 3-space-dimensions NLS equation is not integrable either classically or quantum mechanically. But a number of recently discovered closely related lattices in one space dimension (one space plus one time dimension) are integrable as both classical lattices and quantum lattices while their continuum limits are the now well-known fundamental and integrable system the quantum ‘Bose gas’. These models are all examined in this paper in a physical application of recent so-called ‘quantum groups’ theory, itself fundamental to integrability theory. The ‘superfluid’ phase transitions shown by these lattices, as well as by the bose gas, all at zero temperature in 1 + 1 dimensions, are analysed in terms of the behaviour of certain lattice correlation functions which are either quantum or, in the case of the so-called XY-model, classical correlation functions. Although the repulsive NLS models in 1 + 1 are integrable, they do not have actual soliton solutions. Nevertheless the material as surveyed here is a fundamental application of soliton-theory in the broader context of integrability or near-integrability which has had profound effects in the evolution of current understandings in all of modern theoretical physics. |
