Stability of Leapfrogging Vortex Pairs: A Semi-analytic Approach
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.
Brandon defended his dissertation on leapfrogging point vortices in August 2020. He spent about a year as a postdoc in the Dynamical Systems Laboratory at the NYU Tandon School of Engineering. He is now a data scientist at Prudential Insurance, and has a very cute baby girl.
Brandon completed his Ph.D. in Mathematical Sciences at NJIT. His dissertation project concerns the stability and associated dynamics of systems of leapfrogging vortices.
I am a professor in the Department of Mathematical Sciences at NJIT. My research applies methods of dynamical systems to (conservative, dispersive) nonlinear wave equations arising, for example, in optics and fluid mechanics. I am interested in dynamical and bifurcation phenomena, and I use techniques from Hamiltonian mechanics and perturbation theory, as well as extensive computations. My current research in vortex dynamics is supported by the NSF under DMS-2206016.
I led the effort to start the SIAM New York-New Jersey-Pennsylvania Section (SIAM NNP) and was its first president in 2023-2024. I have handed over the reins to a new leadership group.