Recently a nonlinear Schr\"odinger equation (NLSE) with an inhomogeneous term proportional to $b\mathrm{ln}({|\ensuremath{\psi}|}^{2}|{a}^{3})\ensuremath{\psi}$ has been put forward. It has been proposed to apply it to atomic physics. Subsequent neutron interferometer experiments designed to test the physical reality of such a nonlinearity were not conclusive, thus rejecting it as unphysical. In the present paper it is pointed out that the different length scales
a associated with atomic and nuclear physics, for example, lead to different typical energies
b for these systems. Guided by the experience with phenomenological NLSE's, the constant
b is for the following applications to nuclear physics identified with the compressibility of finite nuclear matter,
C=9K, i.e., $b\ensuremath{\equiv}C$. Thus we obtain consistent qualitative and quantitative answers related to the concepts of microworlds and mesoworlds as well as, e.g., the prediction
130< K< 250 MeV. However, this necessitates the interpretation of the respective NLSE as an equation for extended objects.
We present numerical results from solving the time-dependent nonlinear Schr\"odinger equation (NLSE) that describes an inhomogeneous, weakly interacting Bose-Einstein condensate in a small harmonic trap potential at zero temperature. With this method we are able to find solutions for the NLSE for ground state condensate wave functions in one dimension or in three dimensions with spherical symmetry. These solutions corroborate previous ground state results obtained from the solution of the time-independent NLSE. Furthrmore, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering lengths in one-dimensional (1D) and 3D systems for a fixed number of atoms. In both the 1D and 3D cases, these negative scattering length solutions have solitonlike properties. In 3D, however, these solutions are only stable for a modest range of nonlinearities. We analyze the prospects for diagnosing Bose-Einstein condensation in a trap using several experiments that exploit the time-dependent behavior of the condensate.
Numerical solution of nonlinear Schrodinger equation (NLSE) can be used to describe an inhomogeneous,weakly interact Bose-Einstein condensate in a small harmonic trap potential at zero temperature. In one dimension situation, we find solutions for the NLSE for ground-state condensate wave function, and obtain the ground-state condensate energy per particle. Furthermore, with the time-dependent NLSE, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering length in one-dimensional (1D) system for a fixed number of atoms, and the negative scattering length solutions have solitonlike properties.
By employing supersymmetric quantum mechanics, we present a general algorithm to construct supersymmetric partner potentials and hence derive exact stationary solutions of the inhomogeneous nonlinear Schrödinger equation (INLSE). This is possible due to the connection between the INLSE and the nonlinear Schrödinger equation (NLSE), which can be established from a treatment based on Lie point symmetries and is related with Schrödinger equation, under certain conditions. As an illustrative example, we construct exact solutions for the INLSE through a Pösch-Teller potential with a single bound state.