me | Roy Goodman

Phase portraits and the bifurcation set for the three-vortex interaction system

Tue, 22 Apr 2025 21:36:37 -0400

QGLAB: A MATLAB Package for Computations on Quantum Graphs

Mon, 31 Mar 2025 21:57:07 -0500

A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

Thu, 06 Jun 2024 00:00:00 +0000

Errata: The variables $X_i$ and $Y_i$ in equation (19) are not defined in the text. They are the components of the reduced position vectors $\mathbf{R}_i$. (h/t Tomoki Ohsawa)

Apodizer Design to Efficiently Couple Light into a Fiber Bragg Grating

Tue, 06 Jun 2023 09:44:54 -0400

Transition to instability of the leapfrogging vortex quartet

Thu, 09 Feb 2023 13:30:53 -0400

Efficient Manipulation of Bose-Einstein Condensates in a Double-Well Potential

Wed, 08 Jun 2022 08:43:35 -0400

A Reduction-Based Strategy for Optimal Control of Bose-Einstein Condensates

Mon, 28 Feb 2022 09:35:18 -0500

An Optimal Control Approach to Gradient-Index Design for Beam Reshaping

Thu, 02 Dec 2021 09:35:30 -0500

Solitary Waves in Mass-in-Mass Lattices

Wed, 04 Nov 2020 18:46:43 -0500

Loss of Physical Reversibility in Reversible Systems

Fri, 17 Apr 2020 00:00:00 +0000

Transfer entropy for network reconstruction in a simple dynamical model

Fri, 14 Feb 2020 11:40:00 -0500

Stability of Leapfrogging Vortex Pairs: A Semi-analytic Approach

Thu, 26 Dec 2019 00:00:00 +0000

Erratum

The matrix in Eq. (19) should read $$\small{\left.A_{h}(\theta)\right\rvert_{h=\frac18}= \begin{pmatrix} -\frac{4 \sin {2\theta}}{\sqrt{8 \cos {2\theta}+17}} & \frac{8 \cos ^2{2\theta}-12 \cos {2\theta}+3 \sqrt{8 \cos {2\theta}+17}-11}{2 (1-\cos {2\theta}) \sqrt{8 \cos {2\theta}+17}} \\ \frac{-8 \cos^2{2\theta}-4 \cos {2\theta}-\sqrt{8 \cos {2\theta}+17}+7}{2 (\cos {2\theta}+1) \sqrt{8 \cos {2\theta}+17}} & \frac{4 \sin{2 \theta}}{\sqrt{8 \cos {2\theta}+17}} \end{pmatrix} } $$ This is an isolated error in the writing and does not effect any of the mathematics in the paper.

Addendum

In the published paper, we show that if the differential equation $$ \frac{{\rm d}}{{\rm d}\theta} \begin{pmatrix} \xi_- \\ \eta_+ \end{pmatrix} = A_h(\theta) \begin{pmatrix} \xi_- \\ \eta_+ \end{pmatrix} $$ has a periodic orbit at $h=\frac18$ of period $\pi$, then the leapfrogging orbit bifurcates at $h=\frac18$. We tried using Mathematica to solve this equation exactly but it failed to return a closed form solution. Therefore in the paper we used high-precision numerics and a high-order perturbation expansion to give strong evidence that the solution with initial condition $(1,0)$ is indeed periodic.

In the summer of 2020 we posted a question to the website MathOverflow asking if anyone could find an exact solution to this equation. Almost immediately, we received the solution from Robert Israel, who found the answer using Maple: $$ \begin{aligned} \xi_- (\theta)&= \phantom{-}\frac{1}{20}\left( 1+4 \cos 2\theta+3\sqrt{17+8 \cos 2\theta} \right) \\ \eta_+(\theta)&= -\frac{\tan\theta}{20}\left( 1+4 \cos 2\theta+\sqrt{17+8 \cos 2\theta} \right) . \end{aligned} $$ Thus, without recourse to perturbation theory or numerics, this exact solution proves the theorem.

Drift of spectrally stable shifted states on star graphs

Thu, 31 Oct 2019 00:00:00 +0000

Topological features determining the error in the inference of networks using transfer entropy

Thu, 31 Oct 2019 00:00:00 +0000

Mathematical Analysis of Fractal Kink-Antikink Collisions in the $\varphi^4$ Model

Tue, 01 Jan 2019 00:00:00 +0000

NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

Tue, 01 Jan 2019 00:00:00 +0000

What do I do with all these numerical simulations?

Sat, 14 Apr 2018 11:40:00 -0500

Bifurcations of relative periodic orbits in NLS/GP with a triple-well potential

Sun, 01 Jan 2017 00:00:00 +0000

Complex Behavior in Coupled Nonlinear Waveguides

Tue, 01 Nov 2016 14:00:00 -0600

A Mechanical Analog of the Two-Bounce Resonance of Solitary Waves: Modeling and Experiment

Thu, 01 Jan 2015 00:00:00 +0000

For more about NJIT’s applied math capstone class, see here.

Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates

Thu, 01 Jan 2015 00:00:00 +0000

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

Thu, 01 Jan 2015 00:00:00 +0000

High-order Adaptive Method for Computing Two-dimensional Invariant Manifolds of Maps

Tue, 01 Jan 2013 00:00:00 +0000

Part 2 of Jacek Wróbel’s dissertation.

Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes

Sat, 01 Jan 2011 00:00:00 +0000

Erratum: At the end of section 2.4.3, top of page 10, I state that “a straightforward calculation” shows that the HH bifurcation studied by Johansson for the periodic NLS trimer to come from a semisimple -1:1 resonance. In fact, it is the generic non-semisimple bifurcation.

High-Order Bisection Method for Computing Invariant Manifolds of Two-Dimensional Maps

Sat, 01 Jan 2011 00:00:00 +0000

Part 1 of Jacek Wróbel’s dissertation.

Nonlinear hydrodynamic phenomena in Stokes flow regime

Fri, 01 Jan 2010 00:00:00 +0000

Chaotic scattering in solitary wave interactions: A singular iterated-map description

Tue, 01 Jan 2008 00:00:00 +0000

Hysteretic and chaotic dynamics of viscous drops in creeping flows with rotation

Tue, 01 Jan 2008 00:00:00 +0000

Stability and instability of nonlinear defect states in the coupled mode equations---analytical and numerical study

Tue, 01 Jan 2008 00:00:00 +0000

Chaotic Scattering and the $n$-bounce Resonance in Solitary Wave Interactions

Mon, 01 Jan 2007 00:00:00 +0000

Kink-antikink collisions in the $\phi^4$ equation: The $n$-bounce resonance and the separatrix map

Sat, 01 Jan 2005 00:00:00 +0000

Trapping light with grating defects

Sat, 01 Jan 2005 00:00:00 +0000

Vector soliton interactions in birefringent optical fibers

Sat, 01 Jan 2005 00:00:00 +0000

Interaction of sine-Gordon kinks with defects: The two-bounce resonance

Thu, 01 Jan 2004 00:00:00 +0000

Strong NLS soliton--defect interactions

Thu, 01 Jan 2004 00:00:00 +0000

Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model

Tue, 01 Jan 2002 00:00:00 +0000

Stopping Light on a Defect

Tue, 01 Jan 2002 00:00:00 +0000

Trapping of kinks and solitons by defects: phase space transport in finite-dimensional models

Tue, 01 Jan 2002 00:00:00 +0000

Modulations in the leading edges of midlatitude storm tracks

Mon, 01 Jan 2001 00:00:00 +0000

Nonlinear propagation of light in one-dimensional periodic structures

Mon, 01 Jan 2001 00:00:00 +0000

Trigger Waves in a Model for Catalysis

Sun, 01 Jan 1995 00:00:00 +0000