Jeremy L. Marzuola

Fast Soliton Scattering by Delta Impurities

We study the Gross-Pitaevskii equation with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons.

Strichartz Estimates on Schwarzschild Black Hole Backgrounds

We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of [29], to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.

Soliton Splitting by External Delta Potentials

We show that in the dynamics of the nonlinear Schrodinger equation a soliton scattered by an external delta potential splits into two solitons and a radiation term. Theoretical analysis gives the amplitudes and phases of the reflected and transmitted solitons with errors going to zero as the velocity of the incoming soliton tends to infinity. Numerical analysis shows that this asymptotic relation is valid for all but very slow solitons. We also show that the total transmitted mass, that is, the square of the L2 norm of the solution restricted on the transmitted side of the delta potential, is in good agreement with the quantum transmission rate of the delta potential.

Existence and stability of solitons for the nonlinear Schrödinger equation on hyperbolic space

We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.

Spectral analysis for matrix Hamiltonian operators

In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schrodinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Although we focus on a proof of the 3D cubic problem, this work presents a new algorithm for verifying certain spectral properties needed to study soliton stability.Source code for verification of our computations, and for further experimentation, is available at http://hdl.handle.net/1807/25174.

Wave operator bounds for one-dimensional Schrödinger operators with singular potentials and applications

Boundedness of wave operators for Schrodinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive estimates and commutator bounds.

Quasilinear Schrödinger equations II: Small data and cubic nonlinearities

Author(s): Marzuola, Jeremy L; Metcalfe, Jason; Tataru, Daniel | Abstract: In part I of this project we examined low regularity local well-posedness for generic quasilinear Schr quot;odinger equations with small data. This improved, in the small data regime, the preceding results of Kenig, Ponce, and Vega as well as Kenig, Ponce, Rolvung, and Vega. In the setting of quadratic interactions, the (translation invariant) function spaces which were utilized incorporated an

A Morse Index Theorem for Elliptic Operators on Bounded Domains

l^1

Nonlinear Bound States on Weakly Homogeneous Spaces

summability over cubes in order to account for Mizohatas integrability condition, which is a necessary condition for the

Wave packet parametrices for evolutions governed by pdo’s with rough symbols

L^2