Analysis of 2+1 diffusive-dispersive PDE arising in river braiding

We present local existence and uniqueness results for the following $2+1$ dispersive diffusive equation due to P. Hall arising in modeling of river braiding: $$u_{yyt} - \gamma u_{xxx} -\alpha u_{yyyy} - \beta u_{yy} + \left (u^2 \right)_{xyy} = 0$$ for $(x,y) \in [0, 2\pi] \times [0, \pi]$, $t> 0$, with boundary condition $u_{y}=0=u_{yyy}$ at $y=0$ and $y=\pi$ and $2\pi$ periodicity in $x$, using a contraction mapping argument in a Bourgain-type space $T_{s,b}$. We also show that the energy $\| u \|^2_{L^2}$ and cumulative dissipation $\int_0^t \| u_y \|_{L^2}^2 dt$ are globally controlled in time.Comment: 21 page

An analytical model for the detection of levitated nanoparticles in optomechanics

Interferometric position detection of levitated particles is crucial for the centre-of-mass (CM) motion cooling and manipulation of levitated particles. In combination with balanced detection and feedback cooling, this system has provided picometer scale position sensitivity, zeptonewton force detection, and sub-millikelvin CM temperatures. In this article, we develop an analytical model of this detection system and compare its performance with experimental results allowing us to explain the presence of spurious frequencies in the spectra

Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity

In this paper we prove the breakdown of an heteroclinic connection in the analytic versal unfoldings of the generic Hopf-Zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order if one performs the normal form around the origin, therefore it is a phenomenon "beyond all orders". In this paper we provide a formula for the distance between the corresponding stable and unstable one dimensional manifolds which is given by an exponentially small function in the perturbation parameter. Our result applies both for conservative and dissipative unfoldings

Adjustable microchip ring trap for cold atoms and molecules

We describe the design and function of a circular magnetic waveguide produced from wires on a microchip for atom interferometry using deBroglie waves. The guide is a two-dimensional magnetic minimum for trapping weak-field seeking states of atoms or molecules with a magnetic dipole moment. The design consists of seven circular wires sharing a common radius. We describe the design, the time-dependent currents of the wires and show that it is possible to form a circular waveguide with adjustable height and gradient while minimizing perturbation resulting from leads or wire crossings. This maximal area geometry is suited for rotation sensing with atom interferometry via the Sagnac effect using either cold atoms, molecules and Bose-condensed systems

Whirling skirts and rotating cones

Steady, dihedrally symmetric patterns with sharp peaks may be observed on a spinning skirt, lagging behind the material flow of the fabric. These qualitative features are captured with a minimal model of traveling waves on an inextensible, flexible, generalized-conical sheet rotating about a fixed axis. Conservation laws are used to reduce the dynamics to a quadrature describing a particle in a three-parameter family of potentials. One parameter is associated with the stress in the sheet, aNoether is the current associated with rotational invariance, and the third is a Rossby number which indicates the relative strength of Coriolis forces. Solutions are quantized by enforcing a topology appropriate to a skirt and a particular choice of dihedral symmetry. A perturbative analysis of nearly axisymmetric cones shows that Coriolis effects are essential in establishing skirt-like solutions. Fully non-linear solutions with three-fold symmetry are presented which bear a suggestive resemblance to the observed patterns.Comment: two additional figures, changes to text throughout. journal version will have a wordier abstrac