A System of ODEs for a Perturbation of a Minimal Mass Soliton

We study soliton solutions to the nonlinear Schrödinger equation (NLS) with a saturated nonlinearity. NLS 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 extending the work of Comech and Pelinovsky (Commun. Pure Appl. Math. 56:1565–1607, 2003), which models the behavior of the perturbation for short times. We then provide numerical evidence that under this system of ODEs there are two possible dynamical outcomes, in accord with the conclusions of Pelinovsky et al. (Phys. Rev. E 53(2):1940–1953, 1996). Generically, initial data which supports a soliton structure appears to oscillate, with oscillations centered on a stable soliton. For initial data which is expected to disperse, the finite dimensional dynamics initially follow the unstable portion of the soliton curve.     

Nodal deficiency,spectral flow,and the Dirichlet-to-Neumann map

Letters in Mathematical Physics - It has been recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the...     

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 (L ² 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.     

Nonlinear Bound States on Weakly Homogeneous Spaces

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call F _λ-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.     

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 in the small data regime 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.     

A symplectic perspective on constrained eigenvalue problems

The Maslov index is a powerful tool for computing spectra of selfadjoint, elliptic boundary value problems. This is done by counting intersections of a fixed Lagrangian subspace, which designates the boundary conditions, with the set of Cauchy data for the differential operator. We apply this methodology to constrained eigenvalue problems, in which the operator is restricted to a (not necessarily invariant) subspace. The Maslov index is defined and used to compute the Morse index of the constrained operator. We then prove a constrained Morse index theorem, which says that the Morse index of the constrained problem equals the number of constrained conjugate points, counted with multiplicity, and give an application to the nonlinear Schrödinger equation.     

Eigenfunctions for Partially Rectangular Billiards

In this note, we further develop the methods of Burq and Zworski (2005 Burq , N. , Zworski , M. ( 2005 ). Bouncing ball modes and quantum chaos . SIAM Review 47 ( 5 ): 43 – 49 [CROSSREF] [CSA] [Crossref] , [Google Scholar]) to study eigenfunctions for billiards which have rectangular components: these include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards (Bogomolny et al., 1999 Bogomolny , E. , Gerland , U. , Schmit , C. ( 1999 ). Models of intermediate spectral statistics . Phys. Rev. E 59 : 1315 – 1318 [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). The results are an application of a “black-box” point of view as presented in Burq and Zworski (2004 Burq , N. , Zworski , M. ( 2004 ). Geometric control in the presence of a black box . JAMS 17 : 443 – 471 [CROSSREF] [CSA] [Web of Science ®] , [Google Scholar]).     

Bose–Einstein condensation transition studies for atoms confined in Laguerre–Gaussian laser modes

Multiply-connected traps for cold, neutral atoms fix vortex cores of quantum gases. Laguerre–Gaussian laser modes are ideal for such traps due to their phase stability. We report theoretical calculations of the Bose–Einstein condensation transition properties and thermal characteristics of neutral atoms trapped in multiply connected geometries formed by Laguerre–Gaussian (LG_p^l) beams. Specifically, we consider atoms confined to the anti-node of a LG₀¹ laser mode detuned to the red of an atomic resonance frequency, and those confined in the node of a blue-detuned LG₁¹ beam. We compare the results of using the full potential to those approximating the potential minimum with a simple harmonic oscillator potential. We find that deviations between calculations of the full potential and the simple harmonic oscillator can be up to 3%–8% for trap parameters consistent with typical experiments.