A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

We consider the problem of identifying sharp criteria under which radial $H^1$ (finite energy) solutions to the focusing 3d cubic nonlinear Schr\"odinger equation (NLS) $i\partial_t u + \Delta u + |u|^2u=0$ scatter, i.e. approach the solution to a linear Schr\"odinger equation as $t\to \pm \infty$. The criteria is expressed in terms of the scale-invariant quantities $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}$ and $M[u]E[u]$, where $u_0$ denotes the initial data, and $M[u]$ and $E[u]$ denote the (conserved in time) mass and energy of the corresponding solution $u(t)$. The focusing NLS possesses a soliton solution $e^{it}Q(x)$, where $Q$ is the ground-state solution to a nonlinear elliptic equation, and we prove that if $M[u]E[u]<M[Q]E[Q]$ and $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} < \|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then the solution $u(t)$ is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution $e^{it}Q(x)$, for which equality in these conditions is obtained, is global but does not scatter. We further show that if $M[u]E[u] \|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle \cite{KM06a} in their study of the energy-critical NLS

Kaon-Nucleon Scattering Amplitudes and Z∗^*-Enhancements from Quark Born Diagrams

We derive closed form kaon-nucleon scattering amplitudes using the ``quark Born diagram" formalism, which describes the scattering as a single interaction (here the OGE spin-spin term) followed by quark line rearrangement. The low energy I=0 and I=1 S-wave KN phase shifts are in reasonably good agreement with experiment given conventional quark model parameters. For $k_{lab}> 0.7$ Gev however the I=1 elastic phase shift is larger than predicted by Gaussian wavefunctions, and we suggest possible reasons for this discrepancy. Equivalent low energy KN potentials for S-wave scattering are also derived. Finally we consider OGE forces in the related channels K$\Delta$, K$^*$N and K$^*\Delta$, and determine which have attractive interactions and might therefore exhibit strong threshold enhancements or ``Z$^*$-molecule" meson-baryon bound states. We find that the minimum-spin, minimum-isospin channels and two additional K$^*\Delta$ channels are most conducive to the formation of bound states. Related interesting topics for future experimental and theoretical studies of KN interactions are also discussed.Comment: 34 pages, figures available from the authors, revte

Surface Effects in Magnetic Microtraps

We have investigated Bose-Einstein condensates and ultra cold atoms in the vicinity of a surface of a magnetic microtrap. The atoms are prepared along copper conductors at distances to the surface between 300 um and 20 um. In this range, the lifetime decreases from 20 s to 0.7 s showing a linear dependence on the distance to the surface. The atoms manifest a weak thermal coupling to the surface, with measured heating rates remaining below 500 nK/s. In addition, we observe a periodic fragmentation of the condensate and thermal clouds when the surface is approached.Comment: 4 pages, 4 figures; v2: corrected references; v3: final versio

A fundamental limit for integrated atom optics with Bose-Einstein condensates

The dynamical response of an atomic Bose-Einstein condensate manipulated by an integrated atom optics device such as a microtrap or a microfabricated waveguide is studied. We show that when the miniaturization of the device enforces a sufficiently high condensate density, three-body interactions lead to a spatial modulational instability that results in a fundamental limit on the coherent manipulation of Bose-Einstein condensates.Comment: 6 pages, 3 figure

Continuous loading of a magnetic trap

We have realized a scheme for continuous loading of a magnetic trap (MT). ^{52}Cr atoms are continuously captured and cooled in a magneto-optical trap (MOT). Optical pumping to a metastable state decouples atoms from the cooling light. Due to their high magnetic moment (6 Bohr magnetons), low-field seeking metastable atoms are trapped in the magnetic quadrupole field provided by the MOT. Limited by inelastic collisions between atoms in the MOT and in the MT, we load 10^8 metastable atoms at a rate of 10^8 atoms/s below 100 microkelvin into the MT. After loading we can perform optical repumping to realize a MT of ground state chromium atoms.Comment: 4 pages, 4 figures, version 2, modified references, included additional detailed information, minor changes in figure 3 and in tex

Dynamical Systems approach to Saffman-Taylor fingering. A Dynamical Solvability Scenario

A dynamical systems approach to competition of Saffman-Taylor fingers in a channel is developed. This is based on the global study of the phase space structure of the low-dimensional ODE's defined by the classes of exact solutions of the problem without surface tension. Some simple examples are studied in detail, and general proofs concerning properties of fixed points and existence of finite-time singularities for broad classes of solutions are given. The existence of a continuum of multifinger fixed points and its dynamical implications are discussed. The main conclusion is that exact zero-surface tension solutions taken in a global sense as families of trajectories in phase space spanning a sufficiently large set of initial conditions, are unphysical because the multifinger fixed points are nonhyperbolic, and an unfolding of them does not exist within the same class of solutions. Hyperbolicity (saddle-point structure) of the multifinger fixed points is argued to be essential to the physically correct qualitative description of finger competition. The restoring of hyperbolicity by surface tension is discussed as the key point for a generic Dynamical Solvability Scenario which is proposed for a general context of interfacial pattern selection.Comment: 3 figures added, major rewriting of some sections, submitted to Phys. Rev.

Stability of trapped Bose-Einstein condensates

In three-dimensional trapped Bose-Einstein condensate (BEC), described by the time-dependent Gross-Pitaevskii-Ginzburg equation, we study the effect of initial conditions on stability using a Gaussian variational approach and exact numerical simulations. We also discuss the validity of the criterion for stability suggested by Vakhitov and Kolokolov. The maximum initial chirp (initial focusing defocusing of cloud) that can lead a stable condensate to collapse even before the number of atoms reaches its critical limit is obtained for several specific cases. When we consider two- and three-body nonlinear terms, with negative cubic and positive quintic terms, we have the conditions for the existence of two phases in the condensate. In this case, the magnitude of the oscillations between the two phases are studied considering sufficient large initial chirps. The occurrence of collapse in a BEC with repulsive two-body interaction is also shown to be possible.Comment: 15 pages, 11 figure

Orbital stability: analysis meets geometry

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system

Atomic diffraction from nanostructured optical potentials

We develop a versatile theoretical approach to the study of cold-atom diffractive scattering from light-field gratings by combining calculations of the optical near-field, generated by evanescent waves close to the surface of periodic nanostructured arrays, together with advanced atom wavepacket propagation on this optical potential.Comment: 8 figures, 10 pages, submitted to Phys. Rev.