Recent work in mathematical physics and nonlinear optics has shown that Hamiltonians that are non-Hermitian but still symmetric under parity and time reversal can describe eigenstates of a system with real eigenvalues. Other research has also showed that the nonlinear Schrodinger equation can be generalized to describe PT-symmetric systems, which generates novel solutions not described by its Hermitian equivalent. The Hermitian form of the nonlinear Schroedinger equation can also be extended to describe a particular case of the general PT-symmetric NLS, suggesting a connection between the two. I attempted to generate a unitary operator that will be useful for unitary quantum algorithms describing a coupled set of nonlinear Schroedinger equations and the PT-symmetric version of the NLS.
It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be different from 3, it is desirable to analyze what happens in other spatial dimensions
D. It turns out that while for
D≥3 we still get only the Schroedinger's equation, for
D=2, we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for
D=1, we also get a new nonlinear version of the Schroedinger equation.
One-dimensional (1D) Nonlinear Schroedinger Equaation (NLS) provides a good approximation to attractive Bose-Einshtein condensate (BEC) in a quasi 1D cigar-shaped optical trap in certain regimes. 1D NLS is an integrable equation that can be solved through the inverse scattering method. Our observation is that in many cases the parameters of the BEC correspond to the semiclassical (zero dispersion) limit of the focusing NLS. Hence, recent results about the strong asymptotics of the semiclassical limit solutions can be used to describe some interesting phenomena of the attractive 1D BEC. In general, the semiclassical limit of the focusing NLS exibits very strong modulation instability. However, in the case of an analytical initial data, the NLS evolution does displays some ordered structure, that can describe, for example, the bright soliton phenomenon. We discuss some general features of the semiclassical NLS evolution and propose some new observables.