An Adiabatic Invariant Approach to Transverse Instability: Landau Dynamics of Soliton Filaments

Assume a lower-dimensional solitonic structure embedded in a higher dimensional space, e.g., a 1D dark soliton embedded in 2D space, a ring dark soliton in 2D space, a spherical shell soliton in 3D space etc. By extending the Landau dynamics approach [Phys. Rev. Lett. {\bf 93}, 240403 (2004)], we show that it is possible to capture the transverse dynamical modes (the "Kelvin modes") of the undulation of this "soliton filament" within the higher dimensional space. These are the transverse stability/instability modes and are the ones potentially responsible for the breakup of the soliton into structures such as vortices, vortex rings etc. We present the theory and case examples in 2D and 3D, corroborating the results by numerical stability and dynamical computations.Comment: 5 pages, 3 figure

Observation of the Presuperfluid Regime in a Two-Dimensional Bose Gas

In complementary images of coordinate-space and momentum-space density in a trapped 2D Bose gas, we observe the emergence of pre-superfluid behavior. As phase-space density $\rho$ increases toward degenerate values, we observe a gradual divergence of the compressibility $\kappa$ from the value predicted by a bare-atom model, $\kappa_{ba}$. $\kappa/\kappa_{ba}$ grows to 1.7 before $\rho$ reaches the value for which we observe the sudden emergence of a spike at $p=0$ in momentum space. Momentum-space images are acquired by means of a 2D focusing technique. Our data represent the first observation of non-meanfield physics in the pre-superfluid but degenerate 2D Bose gas.Comment: Replace with the version appeared in PR

Extended 2d generalized dilaton gravity theories

We show that an anomaly-free description of matter in (1+1) dimensions requires a deformation of the 2d relativity principle, which introduces a non-trivial center in the 2d Poincare algebra. Then we work out the reduced phase-space of the anomaly-free 2d relativistic particle, in order to show that it lives in a noncommutative 2d Minkowski space. Moreover, we build a Gaussian wave packet to show that a Planck length is well-defined in two dimensions. In order to provide a gravitational interpretation for this noncommutativity, we propose to extend the usual 2d generalized dilaton gravity models by a specific Maxwell component, which gauges the extra symmetry associated with the center of the 2d Poincare algebra. In addition, we show that this extension is a high energy correction to the unextended dilaton theories that can affect the topology of space-time. Further, we couple a test particle to the general extended dilaton models with the purpose of showing that they predict a noncommutativity in curved space-time, which is locally described by a Moyal star product in the low energy limit. We also conjecture a probable generalization of this result, which provides a strong evidence that the noncommutativity is described by a certain star product which is not of the Moyal type at high energies. Finally, we prove that the extended dilaton theories can be formulated as Poisson-Sigma models based on a nonlinear deformation of the extended Poincare algebra.Comment: 21 pages, IOP LaTeX2e preprint classfile, Improved discussions, Minor corrections, More didactic, More self-contained, New results concerning noncommutativity in curved space-time, Accepted for publication in Classical and Quantum Gravity on 02 Jul 200

Proper time is stochastic time in 2d quantum gravity

We show that proper time, when defined in the quantum theory of 2d gravity, becomes identical to the stochastic time associated with the stochastic quantization of space. This observation was first made by Kawai and collaborators in the context of 2d Euclidean quantum gravity, but the relation is even simpler and more transparent in he context of 2d gravity formulated in the framework of CDT (causal dynamical triangulations).Comment: 30 pages, Talk presented at the meeting "Foundations of Space and Time", Cape Town, 10-14 August 2009. To appear in the proceedings, CU

Maximizers for the Strichartz norm for small solutions of mass-critical NLS

Consider the mass-critical nonlinear Schr\"odinger equations in both focusing and defocusing cases for initial data in $L^2$ in space dimension N. By Strichartz inequality, solutions to the corresponding linear problem belong to a global $L^p$ space in the time and space variables, where $p=2+4/N$. In 1D and 2D, the best constant for the Strichartz inequality was computed by D.~Foschi who has also shown that the maximizers are the solutions with Gaussian initial data. Solutions to the nonlinear problem with small initial data in $L^2$ are globally defined and belong to the same global $L^p$ space. In this work we show that the maximum of the $L^p$ norm is attained for a given small mass. In addition, in 1D and 2D, we show that the maximizer is unique and obtain a precise estimate of the maximum. In order to prove this we show that the maximum for the linear problem in 1D and 2D is nondegenerated.Comment: To be published in Annali della Scuola Normale Superiore di Pis