Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data

We compute the scattering data of the Benjamin-Ono equation for arbitrary rational initial conditions with simple poles. Specifically, we obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient.Comment: 16 Pages, 2 Figure

Three Stratified Fluid Models: Benjamin-Ono, Tidal Resonance, and Quasi-Geostrophy.

In this dissertation, we study three models arising in stratified fluid dynamics: the Benjamin-Ono equation, a two-layer tidal model, and the quasi-geostrophic equations. First, we compute the scattering data of the Benjamin-Ono equation for arbitrary rational initial conditions with simple poles, under mild restrictions. For this class of initial conditions, we are able to obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient. We proceed to use these exact formulas to deduce precise asymptotics for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant on the eigenvalue. Second, we present a simplified analytical tidal model that demonstrates the influence of stratification on both large- and small-scale surface tidal elevations in a qualitatively similar manner as in previous global realistic-domain numerical simulations. Our analytical results demonstrate the potential for the presence of stratification and changes to stratification to alter the large-scale (barotropic) tide. We find that changes in stratification may change large-scale tidal elevations at the same order as small-scale tidal elevations if there is significant bottom topography and the damping acts primarily on the bottom layer. Third, we investigate the influence of bottom friction on quasi-geostrophic turbulence dynamics in a multi-level forced-dissipated model that includes buoyancy advection at the boundaries. Motivated by earlier studies of two-layer quasi-geostrophic turbulence, we study the influence of bottom friction on the horizontal scales and vertical structure of eddy kinetic energy with the aid of a ‘surface-aware’ modal decomposition.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/116662/1/wreagan_1.pd

Measurement of charged particle spectra in minimum-bias events from proton-proton collisions at root s =13 TeV

Pseudorapidity, transverse momentum, and multiplicity distributions are measured in the pseudorapidity range vertical bar eta vertical bar 0.5 GeV in proton-proton collisions at a center-of-mass energy of root s = 13 TeV. Measurements are presented in three different event categories. The most inclusive of the categories corresponds to an inelastic pp data set, while the other two categories are exclusive subsets of the inelastic sample that are either enhanced or depleted in single diffractive dissociation events. The measurements are compared to predictions from Monte Carlo event generators used to describe high-energy hadronic interactions in collider and cosmic-ray physics.Peer reviewe