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Jeremy L. Marzuola

Publications and source records attributed to Jeremy L. Marzuola.

86 records · Page 5Linked to original sources
arXiv2013

Dynamics near a minimal-mass soliton for a Korteweg-de Vries equation

We study soliton solutions to a generalized Korteweg - de Vries (KdV) equation with a saturated nonlinearity, following the line of inquiry of the authors for the nonlinear Schrödinger equation (NLS). KdV with such a nonlinearity is known to possess a minimal-mass soliton. We consider a small perturbation of a minimal-mass soliton and identify a system of ODEs, which models the behavior of the perturbation for short times. This connects nicely to a work of Comech, Cuccagna & Pelinovsky (2007). These ODEs form a simple dynamical system with a single unstable hyperbolic fixed point with two possible dynamical outcomes. A particular feature of the dynamics are that they are non-oscillatory. This distinguishes the KdV problem from the analogous NLS one.

nlin.PS↗
arXiv2012

On the regularity of the flow map for the gravity-capillary equations

We prove via explicitly constructed initial data that solutions to the gravity-capillary wave system in $\mathbb{R}^3$ representing a 2d air-water interface immediately fails to be $C^3$ with respect to the initial data if the initial data $(h_0, ψ_0) \in H^{s+\frac12} \otimes H^{s}$ for $s<3$. Similar results hold in $\mathbb{R}^2$ domains with a 1d interface. Furthermore, we discuss the illposedness threshold for the pure gravity water wave system.

math.AP↗
arXiv2012

Behavior of a Model Dynamical System with Applications to Weak Turbulence

We experimentally explore solutions to a model Hamiltonian dynamical system derived in Colliander et al., 2012, to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus. Our results include a statistical analysis of the evolution of data with localized amplitudes and random phases, which supports the conjecture that energy cascades are a generic phenomenon. We also identify stationary solutions, periodic solutions in an associated problem and find experimental evidence of hyperbolic behavior. Many of our results rely upon reframing the dynamical system using a hydrodynamic formulation.

math.AP↗
arXiv2012

Quasilinear Schrödinger equations II: Small data and cubic nonlinearities

In part I of this project we examined low regularity local well-posedness for generic quasilinear Schrödinger 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 $l^1$ summability over cubes in order to account for Mizohata's integrability condition, which is a necessary condition for the $L^2$ well-posedness for the linearized equation. For cubic interactions, this integrability condition meshes better with the inherent $L^2$ nature of the Schrödinger equation, and such summability is not required. Thus we are able to prove small data well-posedness in $H^s$ spaces.

math.AP↗
arXiv2012

Quasilinear Schrödinger equations I: Small data and quadratic interactions

In this article we prove local well-posedness in low-regularity Sobolev spaces for general quasilinear Schrödinger equations. These results represent improvements of the pioneering works by Kenig-Ponce-Vega and Kenig-Ponce-Rolvung-Vega, where viscosity methods were used to prove existence of solutions in very high regularity spaces. Our arguments here are purely dispersive. The function spaces in which we show existence are constructed in ways motivated by the results of Mizohata, Ichinose, Doi, and others, including the authors.

math.AP↗
arXiv2011

Nonconcentration in partially rectangular billiards

In specific types of partially rectangular billiards we estimate the mass of an eigenfunction of energy $E$ in the region outside the rectangular set in the high-energy limit. We use the adiabatic ansatz to compare the Dirichlet energy form with a second quadratic form for which separation of variables applies. This allows us to use sharp one-dimensional control estimates and to derive the bound assuming that $E$ is not resonating with the Dirichlet spectrum of the rectangular part.

math.AP↗
arXiv2011

Strichartz estimates for the wave equation on flat cones

We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius $ρ> 0$, the manifold $\mathbb{R}_+ \times \mathbb{R} / 2 πρ\mathbb{Z}$ equipped with the metric $\g(r,θ) = dr^2 + r^2 dθ^2$. Using explicit representations of the solution operator in regions related to flat wave propagation and diffraction by the cone point, we prove dispersive estimates and hence scale invariant Strichartz estimates for the wave equation on flat cones. We then show that this yields corresponding inequalities on wedge domains, polygons, and Euclidean surfaces with conic singularities. This in turn yields well-posedness results for the nonlinear wave equation on such manifolds. Morawetz estimates on the cone are also treated.

math.AP↗
arXiv2011

Nonlinear quasimodes near elliptic periodic geodesics

We consider the nonlinear Schrödinger equation on a compact manifold near an elliptic periodic geodesic. Using a geometric optics construction, we construct quasimodes to a nonlinear stationary problem which are highly localized near the periodic geodesic. We show the nonlinear Schrödinger evolution of such a quasimode remains localized near the geodesic, at least for short times.

math.AP↗
arXiv2011

On instability of some approximate periodic solutions for the full nonlinear Schrödinger equation

Using the Fermi Golden Rule analysis developed in several results by the first author, we prove asymptotic stability of asymmetric nonlinear bound states bifurcating from linear bound states for a quintic nonlinear Schrödinger operator with symmetric potential. This goes in the direction of proving that the approximate periodic solutions of the NLS shadowed in a recent work of the second author with Michael Weinstein do not persist for the full NLS.

math.AP↗
arXiv2010

Small data scattering and soliton stability in $\dot{H}^{-\frac16}$ for the quartic KdV Equation

In this note we prove scattering for perturbations of solitons in the scaling space appropriate for the quartic nonlinearity, namely $\dot{H}^{-\frac16}$. The article relies strongly on refined estimates for a KdV equation linearized at the soliton. In contrast to the work of Tao (2006), we are able to work purely in the scaling space without additional regularity assumptions, allowing us to prove some results on the existence of inverse wave operators.

math.AP↗
arXiv2010

Spectral Analysis for Matrix Hamiltonian Operators

In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schrödinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Though 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 comptuations, and for further experimentation, are available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.

math.AP↗
arXiv2010

Strichartz estimates for the Schrödinger equation on polygonal domains

We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a result of the second author regarding the Schrödinger equation on the Euclidean cone.

math.AP↗
arXiv2009

Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii Equations

We consider a class nonlinear Schrödinger / Gross-Pitaevskii equations (NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double well linear potential. NLS/GP plays a central role in the modeling of nonlinear optical and mean-field quantum many-body phenomena. It is known that there is a critical $L^2$ norm (optical power / particle number) at which there is a symmetry breaking bifurcation of the ground state. We study the rich dynamical behavior near the symmetry breaking point. The source of this behavior in the full Hamiltonian PDE is related to the dynamics of a finite-dimensional Hamiltonian reduction. We derive this reduction, analyze a part of its phase space and prove a {\it shadowing theorem} on the persistence of solutions, with oscillating mass-transport between wells, on very long, but finite, time scales within the full NLS/GP. The infinite time dynamics for NLS/GP are expected to depart, from the finite dimensional reduction, due to resonant coupling of discrete and continuum / radiation modes.

math.AP↗