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Jeremy L. Marzuola

Publications and source records attributed to Jeremy L. Marzuola.

At least 73 records · Page 4Linked to original sources
arXiv2017

Trigonometric integrators for quasilinear wave equations

Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semi-discretization in time with these integrators is shown for a sufficiently regular exact solution. The time integrators are also combined with a Fourier spectral method into a fully discrete scheme, for which error bounds are provided without requiring any CFL-type coupling of the discretization parameters. The proofs of the error bounds are based on energy techniques and on the semiclassical Gårding inequality.

math.NA↗
arXiv2017

Small data global solutions for the Camassa-Choi equations

We consider solutions to the Cauchy problem for an internal-wave model derived by Camassa-Choi in a paper in Journal of Fluid Mechanics (1996). This model is a natural generalization of the Benjamin-Ono and Intermediate Long Wave equations in the case of weak transverse effects. We prove the existence and long-time dynamics of global solutions from small, smooth, spatially localized initial data on $\mathbb{R}^2$.

math.AP↗
arXiv2016

Gross-Pitaevskii vortex motion with critically-scaled inhomogeneities

We study the dynamics of vortices in an inhomogeneous Gross-Pitaevskii equation $i \partial_t u = Δu + {1\over \varepsilon^2} (p_\varepsilon^2(x) - |u|^2)$. For a unique scaling regime $|p_\varepsilon(x) - 1 | = O(|\log \varepsilon|^{-1})$, it is shown that vortices can interact both with the background perturbation and with each other. Results for associated parabolic and elliptic problems are discussed.

math.AP↗
arXiv2016

Existence and Stability Properties of Radial Bound States for Schrödinger-Poisson with an External Coulomb Potential in Three Space Dimensions

We consider radial solutions to the Schrödinger-Poisson system in three dimensions with an external smooth potential with Coulomb-like decay. Such a system can be viewed as a model for the interaction of dark matter with a bright matter background in the non-relativistic limit. We find that there are infinitely many critical points of the Hamiltonian, subject to fixed mass, and that these bifurcate from solutions to the associated linear problem at zero mass. As a result, each branch has a different topological character defined by the number of zeros of the radial states. We construct numerical approximations to these nonlinear states along the first several branches. The solution branches can be continued, numerically, to large mass values, where they become asymptotic, under a rescaling, to those of the Schrödinger-Poisson problem with no external potential. Our numerical computations indicate that the ground state is orbitally stable, while the excited states are linearly unstable for sufficiently large mass.

math.AP↗
arXiv2016

Spectrally Optimized Pointset Configurations

The search for optimal configurations of pointsets, the most notable examples being the problems of Kepler and Thompson, have an extremely rich history with diverse applications in physics, chemistry, communication theory, and scientific computing. In this paper, we introduce and study a new optimality criteria for pointset configurations. Namely, we consider a certain weighted graph associated with a pointset configuration and seek configurations which minimize certain spectral properties of the adjacency matrix or graph Laplacian defined on this graph, subject to geometric constraints on the pointset configuration. This problem can be motivated by solar cell design and swarming models, and we consider several spectral functions with interesting interpretations such as spectral radius, algebraic connectivity, effective resistance, and condition number. We prove that the regular simplex extremizes several spectral invariants on the sphere. We also consider pointset configurations on flat tori via (i) the analogous problem on lattices and (ii) through a variety of computational experiments. For many of the objectives considered (but not all), the triangular lattice is extremal.

math.SP↗
arXiv2016

An Isoperimetric inequality for an integral operator on flat tori

We consider a class of Hilbert-Schmidt integral operators with an isotropic, stationary kernel acting on square integrable functions defined on flat tori. For any fixed kernel which is positive and decreasing, we show that among all unit-volume flat tori, the equilateral torus maximizes the operator norm and the Hilbert-Schmidt norm.

math.SP↗
arXiv2016

$L^p$-bounds on spectral clusters associated to polygonal domains

We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or more generally Euclidean surfaces with conic singularities) by analysis of the wave operator on the flat Euclidean cone $C(\mathbb{S}^1_ρ) := \mathbb{R}_+ \times \left(\mathbb{R} \big/ 2πρ\mathbb{Z}\right)$ of radius $ρ> 0$ equipped with the metric $h(r,θ) = d r^2 + r^2 \, dθ^2$. Using explicit oscillatory integrals and relying on the fundamental solution to the wave equation in geometric regions related to flat wave propagation and diffraction by the cone point, we can prove spectral cluster estimates equivalent to those in works on smooth Riemannian manifolds.

math.AP↗
arXiv2016

Manifold decompositions and indices of Schrödinger operators

The Maslov index is used to compute the spectra of different boundary value problems for Schrödinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold $M$ divided into components $Ω_1$ and $Ω_2$ by a separating hypersurface $Σ$. A homotopy argument relates the spectrum of a second-order elliptic operator on $M$ to its Dirichlet and Neumann spectra on $Ω_1$ and $Ω_2$, with the difference given by the Maslov index of a path of Lagrangian subspaces. This Maslov index can be expressed in terms of the Morse indices of the Dirichlet-to-Neumann maps on $Σ$. Applications are given to doubling constructions, periodic boundary conditions and the counting of nodal domains. In particular, a new proof of Courant's nodal domain theorem is given, with an explicit formula for the nodal deficiency.

math.AP↗
arXiv2015

Euler Equation on a Rotating Surface

We study 2D Euler equations on a rotating surface, subject to the effect of the Coriolis force, with an emphasis on surfaces of revolution. We bring in conservation laws that yield long time estimates on solutions to the Euler equation, and examine ways in which the solutions behave like zonal fields, building on work of B.~Cheng and A.~Mahalov, examining how such 2D Euler equations can account for the observed band structure of rapidly rotating planets. Specific results include both an analysis of time averages of solutions and a study of stability of stationary zonal fields. The latter study includes both analytical and numerical work.

math.AP↗
arXiv2015

A Morse index theorem for elliptic operators on bounded domains

Given a selfadjoint, elliptic operator $L$, one would like to know how the spectrum changes as the spatial domain $Ω\subset \mathbb{R}^d$ is deformed. For a family of domains $\{Ω_t\}_{t\in[a,b]}$ we prove that the Morse index of $L$ on $Ω_a$ differs from the Morse index of $L$ on $Ω_b$ by the Maslov index of a path of Lagrangian subspaces on the boundary of $Ω$. This is particularly useful when $Ω_a$ is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the "original" problem (on $Ω_b$) and the "simplified" problem (on $Ω_a$). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.

math.AP↗
arXiv2015

The Structure of Global Attractors for Dissipative Zakharov Systems with Forcing on the Torus

The Zakharov system was originally proposed to study the propagation of Langmuir waves in an ionized plasma. In this paper, motivated by earlier work of the first and third authors, we numerically and analytically investigate the dynamics of the dissipative Zakharov system on the torus in 1 dimension. We find an interesting family of stable periodic orbits and fixed points, and explore bifurcations of those points as we take weaker and weaker dissipation.

math.AP↗
arXiv2014

Strang Splitting Methods for a quasilinear Schrödinger equation - Convergence, Instability and Dynamics

We study the Strang splitting scheme for quasilinear Schrödinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the numerical blow-up of large data solutions and connects to analytical breakdown of regularity of solutions to quasilinear Schrödinger equations. Numerical tests are performed for a modified version of the superfluid thin film equation.

math.NA↗
arXiv2014

Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii Equation

We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential, continuing work of the 2nd and 3rd authors. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption. The optical power ($L^2$ norm) is conserved with propagation distance. At low optical power, the beam energy executes beating oscillations between the two waveguides. There is an optical power threshold above which the set of guided mode solutions splits into two families of solutions. One type of solution corresponds to an optical beam which is concentrated in either waveguide, but not both. Solutions in the second family undergo tunneling oscillations between the two waveguides. NLS/GP can also model the behavior of Bose-Einstein condensates. A finite dimensional reduction (system of ODEs) well-approximates the PDE dynamics on long time scales. In particular, we derive this reduction, find a class of exact solutions and prove the very long-time shadowing of these solutions by applying the approach of the 2nd and 3rd authors.

math.AP↗
arXiv2013

Higher Dimensional Vortex Standing Waves for Nonlinear Schr{ö}dinger Equations

We study standing wave solutions to nonlinear Schr{ö}dinger equations, on a manifold with a rotational symmetry, which transform in a natural fashion under the group of rotations. We call these vortex solutions. They are higher dimensional versions of vortex standing waves that have been studied on the Euclidean plane. We focus on two types of vortex solutions, which we call spherical vortices and axial vortices.

math.AP↗
arXiv2013

Strichartz estimates on exterior polygonal domains

Using a new local smoothing estimate of the first and third authors, we prove local-in-time Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and global-in-time Strichartz estimates without a loss exterior to a large class of polygonal obstacles with Dirichlet boundary conditions. In addition, we prove a global-in-time local smoothing estimate in exterior wedge domains with Dirichlet boundary conditions and discuss some nonlinear applications.

math.AP↗
arXiv2013

The relaxation of a family of broken bond crystal surface models

We study the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. With computational experiments and theoretical arguments we are able to derive several partial differential equation limits identified (or nearly identified) in previous studies and to clarify the correct choice of surface tension appearing in the PDE and the correct scaling regime giving rise to each PDE. We also provide preliminary computational investigations of a number of interesting qualitative features of the large scale behavior of the models.

math-ph↗