Winter 2025
January 26, 2026, 4:00–5:00 pm, zoom
Speaker: Aikaterini Gkogkou (Tulane University)
Title: Painlevé Universality class for the maximal amplitude solution of the Focusing Nonlinear Schrödinger Equation with randomness
Abstract: In this work, we establish universality results for the $N$-soliton solution of the focusing NLS equation at maximal amplitude. Specifically, we choose the associated normalization constants so that the solution achieves its maximal peak, which, in the large-$N$ limit, satisfies a Painlevé-type equation independently of the distribution of the (random) discrete eigenvalues. We identify two distinct universality classes, determined by the structure of the discrete eigenvalues: the \textit{Painlevé--III} and \textit{Painlevé--V} rogue-wave solutions. In the Painlevé--III case, the eigenvalues take the form $\lambda_j = v_j + i \mu_j$, while for Painlevé--V they satisfy $\lambda_j = -\zeta \, j + v_j + i \mu_j$, with $0 < \zeta < 1$. In both cases, $v_j$ and $\mu_j$ are sub-exponential random variables. Universality can then be summarized as follows: regardless of the specific realizations of the amplitudes and velocities, provided they are sub-exponential random variables and the normalization constants are chosen to maximize the \(N\)-soliton solution, the resulting maximal peak always corresponds to either a Painlevé--III or Painlevé--V rogue-wave profile in the large-$N$ limit.
February 2, 2026, 4:00–5:00 pm, zoom
Speaker: Jacek Szmigielski (University of Saskatchewan)
Title: Camassa-Holm Equations with an internal symmetry
Abstract: In the first part of my talk, I will revisit my work on the scalar Camassa–Holm equation, which will set the stage for the second part. There, I will outline a construction of spinor analogs of the Camassa–Holm equation. In essence, each orthogonal group gives rise to a Camassa–Holm–type equation with intricate internal dynamics. I will motivate this generalization using spectral deformations of the Euler–Bernoulli beam problem, which corresponds to the Clifford algebra on two generators with Minkowski signature. The dynamics of solutions of this Clifford extension are far more intricate than in the scalar case, a contrast I will illustrate with concrete examples. The talk is based on recent joint work with R. Beals and ongoing research with A. Hone and V. Novikov.
February 9, 2026, 4:00–5:00 pm, zoom
Speaker: Thierry Laurens (University of Wisconsin-Madison)
Title: TBA
Abstract: TBA
February 16, 2026, 4:00–5:00 pm, East Hall 1866
Speaker: Kenta Miyahara (Indiana University Indianapolis)
Title: Transition Asymptotics for the Real Solutions of the sinh-Gordon Painlev\'e III
Abstract: We consider solutions of the sinh-Gordon Painlev\'e III equation
\[
u_{xx} + \frac{1}{x} u_x = \sinh u
\]
that are real on \((0, \infty)\). They are parametrized by the monodromy parameter \( p \in \overline{\mathbb C} \), \( |p|>1 \), and an additional real parameter \( s^{\mathbb R} \) when \( p=\infty \). We describe the transition between singular solutions (\( |p|<\infty \)) and smooth solutions (\( p=\infty \)), as \( x \to +\infty \) and \( p \to \infty \) given that \( 2\Im(p)=-s^\R\).
This presentation is based on the ongoing work with Maxim Yattselev.
February 23, 2026, 4:00–5:00 pm, East Hall 1866
Speaker: Matthew Mitchell (University of Central Florida)
Title: TBA
Abstract: TBA
March 9, 2026, 4:00–5:00 pm, zoom
Speaker: Alessandro Arsie (The University of Toledo)
Title: TBA
Abstract: TBA
March 16, 2026, 4:00–5:00 pm, East Hall 1866
Speaker: Deniz Bilman (University of Cincinnati)
Title: TBA
Abstract: TBA
March 30, 2026, 4:00–5:00 pm, East Hall 1866
Speaker: Robert Jenkins (University of Central Florida)
Title: TBA
Abstract: TBA
April 6, 2026, 4:00–5:00 pm, zoom
Speaker: Benjamin Harrop-Griffiths (Georgetown University)
Title: TBA
Abstract: TBA