Publications

We use Jacobi coordinates and a momentum map/Lie-Poisson reduction to derive a system of equations for the three-vortex problem with no coordinate singularities and use it to classify the phase space as the three circulations vary.

We perform an asymptotic analysis of massive point-vortex dynamics in Bose–Einstein condensates in the small-mass limit \(\varepsilon \to 0\). We define two distinguished manifolds in the phase space of the …

We describe QGLAB, a new MATLAB package for analyzing partial differential equations on quantum graphs. The supplementary material contains additional examples and a complete listing of user-callable functions.

We use Jacobi coordinates and Nambu brackets to derive a new representation of the motion of three vortices which, unlike all previous reductions, introduces no singularities into the system. We use this to study the scattering of dipoles by stationary vortices.

A complete English translation of Walter Gröbli’s 1877 Ph.D. Thesis, together with some notes on the process. The work considers the dynamics of point vortices in a two-dimensional inviscid incompressible fluid and derives a number of exact solutions in the cases of three, four and \(2n\) vortices with certain restriction on the vortices’ circulation and the symmetry of the initial configuration.

We use optimal control based on a Galerkin-truncated model to transfer a BEC from one well of a two-well potential to the other.

We use an optimal control framework to efficiently couple light into a fiber Bragg grating, improving a previously-reported 66 percent transmission to 88 percent.

We show via analytical perturbation theory that the leapfrogging vortex quartet becomes unstable exactly at the critical ratio \(\alpha= \phi^{-2}.\)

We address the problem of reshaping light in the Schrödinger optics regime from the perspective of optimal control theory. In technological applications, Schrödinger optics is often used to model a …

Applications of Bose-Einstein Condensates (BEC) often require that the condensate be prepared in a specific complex state. Optimal control is a reliable framework to prepare such a state while avoiding …

We consider the existence of spatially localized traveling wave solutions of the mass-in-mass lattice. Under an anti-resonance condition first discovered by Kevrekidis, Stefanov and Xu, we prove that such solutions …

A dynamical system is said to be reversible if, given an output, the input can always be recovered in a well-posed manner. Nevertheless, we argue that reversible systems that have …

We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as leapfrogging orbits. These solutions, which consists of two pairs of identical yet oppositely-signed …

When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schrödinger (NLS) …

The problem of inferring interactions from observations of individual behavior in networked dynamical systems is ubiquitous in science and engineering. From brain circuits to financial networks, there is a dire …

We analyze the fractal structure seen in kink-antikink collisions of the \(\varphi^4\) equation. The analysis is based on qualitative ordinary differential equation models related to those obtained using the variational …

We consider the bifurcations of standing wave solutions to the nonlinear Schrödinger equation (NLS) posed on a quantum graph consisting of two loops connected by a single edge, the so-called …

The nonlinear Schrödinger/Gross–Pitaevskii (NLS/GP) equation is considered in the presence of three equally-spaced potentials. The problem is reduced to a finite-dimensional Hamiltonian system by a Galerkin truncation. Families of oscillatory …

We describe a simple mechanical system, a ball rolling along a specially-designed landscape, that mimics the dynamics of a well known phenomenon, the two-bounce resonance of solitary wave collisions, that …

We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic trap. We focus on a system of ordinary differential equations describing the vortices’ motion, …

We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides …

An efficient and accurate numerical method is presented for computing invariant manifolds of maps which arise in the study of dynamical systems. A quasi-interpolation method due to Hering-Bertram et al. …

We examine the dynamics of solutions to nonlinear Schrödinger/Gross– Pitaevskii equations that arise due to semisimple indefinite Hamiltonian Hopf bifurcations—the collision of pairs of eigenvalues on the imaginary axis. We …

We describe an efficient and accurate numerical method for computing smooth approximations to invariant manifolds of planar maps, based on geometric modeling ideas from Computer Aided Geometric Design (CAGD). The …

We investigate nonlinear phenomena in dispersed two-phase systems under creeping-flow conditions. We consider nonlinear evolution of a single deformed drop and collective dynamics of arrays of hydrodynamically coupled particles. To …

We derive a family of singular iterated maps—closely related to Poincaré maps—that describe chaotic interactions between colliding solitary waves. The chaotic behavior of such solitary-wave collisions depends on the transfer …

We investigate nonlinear phenomena in dispersed two-phase systems under creeping-flow conditions. We consider nonlinear evolution of a single deformed drop and collective dynamics of arrays of hydrodynamically coupled particles. To …

Coupled backward and forward wave amplitudes of an electromagnetic field propagating in a periodic and nonlinear medium at Bragg resonance are governed by the nonlinear coupled mode equations (NLCME). This …

We present a new and complete analysis of the n-bounce resonance and chaotic scattering in solitary wave collisions. In these phenomena, the speed at which a wave exits a collision …

We provide a detailed mathematical explanation of a phenomenon known as the two-bounce resonance observed in collisions between kink and anti-kink traveling waves of the phi-four equations of mathematical physics. …

Gap solitons are localized traveling waves that exist in Bragg grating optical fibers. We demonstrate a family of grating defects that support linear and nonlinear standing wave modes, and show …

We consider the interactions of two identical, orthogonally polarized vector solitons in a nonlinear optical fiber with two polarization directions, described by a coupled pair of nonlinear Schrödinger equations. We …

A model of soliton–defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown …

We consider the interaction of a nonlinear Schrödinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under …

We study a model derived by Fei et al. [Phys. Rev. A 45 (1992) 6019] of a kink solution to the sine-Gordon equation interacting with an impurity mode. The model …

Downstream development is a term encompassing a variety of effects relating to the propagation of storm systems at midlatitude. We investigate a mechanism behind downstream development and study how wave …

Gap solitons are localized nonlinear coherent states that have been shown both theoretically and experimentally to propagate in periodic structures. Although theory allows for their propagation at any speed v, …

We study models of Fei et al. and of Forinash et al of kinks in the sine-Gordon equation, and solitons in the nonlinear Schrodinger equation interacting with point defects. The …

We consider the nonlinear propagation of light in an optical fiber waveguide as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is assumed to have an index of refraction …

We consider the model of catalysis due to Ziff, Gulari, and Barshad [Phys Rev. Lett. 56, 2553 (1986)] as a pattern formation problem. We find that the model supports …