Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrodinger equation

The two-dimensional cubic nonlinear Schrodinger equation admits a large family of one-dimensional bounded traveling-wave solutions. All such solutions may be written in terms of an amplitude and a phase. Solutions with piecewise constant phase have been well studied previously. Some of these solutions were found to be stable with respect to one-dimensional perturbations. No such solutions are stable with respect to two-dimensional perturbations. Here we consider stability of the larger class of solutions whose phase is dependent on the spatial dimension of the one-dimensional wave form. We study the spectral stability of such nontrivial-phase solutions numerically, using Hill's method. We present evidence which suggests that all such nontrivial-phase solutions are unstable with respect to both one- and two-dimensional perturbations. Instability occurs in all cases: for both the elliptic and hyperbolic nonlinear Schrodinger equations, and in the focusing and defocusing case.Comment: Submitted: 13 pages, 3 figure

Stability of Repulsive Bose-Einstein Condensates in a Periodic Potential

The cubic nonlinear Schr\"odinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schr\"odinger equation nor in the integrable nonlinear Schr\"odinger equation. Their stability is examined using analytic and numerical methods. All trivial-phase stable solutions are deformations of the ground state of the linear Schr\"odinger equation. Our results show that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Physically, this implies stability of states near the Thomas-Fermi limit.Comment: 12 pages, 17 figure

Stability of Attractive Bose-Einstein Condensates in a Periodic Potential

Using a standing light wave trap, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schr\"odinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schr\"odinger equation nor in the integrable nonlinear Schr\"odinger equation. Their stability is examined using analytic and numerical methods. Trivial-phase solutions are experimentally stable provided they have nodes and their density is localized in the troughs of the potential. Stable time-periodic solutions are also examined.Comment: 12 pages, 18 figure

Berry phases for the nonlocal Gross-Pitaevskii equation with a quadratic potential

A countable set of asymptotic space -- localized solutions is constructed by the complex germ method in the adiabatic approximation for the nonstationary Gross -- Pitaevskii equation with nonlocal nonlinearity and a quadratic potential. The asymptotic parameter is 1/T, where $T\gg1$ is the adiabatic evolution time. A generalization of the Berry phase of the linear Schr\"odinger equation is formulated for the Gross-Pitaevskii equation. For the solutions constructed, the Berry phases are found in explicit form.Comment: 13 pages, no figure

The MOLLER Experiment: An Ultra-Precise Measurement of the Weak Mixing Angle Using Møller Scattering

The physics case and an experimental overview of the MOLLER (Measurement Of a Lepton Lepton Electroweak Reaction) experiment at the 12 GeV upgraded Jefferson Lab are presented. A highlight of the Fundamental Symmetries subfield of the 2007 NSAC Long Range Plan was the SLAC E158 measurement of the parity-violating asymmetry $A_{PV}$ in polarized electron-electron (M{\o}ller) scattering. The proposed MOLLER experiment will improve on this result by a factor of five, yielding the most precise measurement of the weak mixing angle at low or high energy anticipated over the next decade. This new result would be sensitive to the interference of the electromagnetic amplitude with new neutral current amplitudes as weak as $\sim 10^{-3}\cdot G_F$ from as yet undiscovered dynamics beyond the Standard Model. The resulting discovery reach is unmatched by any proposed experiment measuring a flavor- and CP-conserving process over the next decade, and yields a unique window to new physics at MeV and multi-TeV scales, complementary to direct searches at high energy colliders such as the Large Hadron Collider (LHC). The experiment takes advantage of the unique opportunity provided by the upgraded electron beam energy, luminosity, and stability at Jefferson Laboratory and the extensive experience accumulated in the community after a round of recent successfully completed parity-violating electron scattering experiment