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: Continuum Calogero--Moser models
Abstract: The focusing CCM equation is a completely integrable PDE that describes a continuum limit of a particle gas interacting pairwise through an inverse-square potential. This system is well-posed below the mass of the soliton, but above this threshold there are solutions that blow up in finite time.
The defocusing model arises as a modulation equation for internal waves in a deep stratified fluid. In this setting, it is a member of a family of systems known as the intermediate nonlinear Schrödinger equations.
In this talk, we will discuss some recent well-posedness results for the continuum Calogero--Moser and intermediate nonlinear Schrödinger equations. This is based on joint works with Andreia Chapouto, Justin Forlano, Rowan Killip, and Monica Visan.
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, zoom
Speaker: Matthew Mitchell (University of Central Florida)
Title: Semiclassical Soliton Ensembles for the Intermediate Long Wave and Korteweg-de Vries Equations
Abstract: Semiclassical soliton ensembles (SSE) in the small dispersion limit are initially coherent collections of many solitons that well-approximate some initial profile. Evolving forward in time, the profile will eventually undergo wave breaking, shedding the solitons and generating a dispersive shock wave. We study this phenomenon for two PDE. The first SSE, for the intermediate long wave equation, is constructed to approximate general smooth Klaus-Shaw initial data. We first conduct a heuristic WKB approximation to determine the approximate scattering data and then rigorously study the inverse scattering problem using the methods of Lax and Levermore. We show the initial condition is recovered in the limit and the solution up until wave breaking approaches that of Invicid Burgers' equation in an L^2 sense. The second SSE is the sech^2 initial condition for the Korteweg-de Vries equation. Inverse scattering is done via a Reimann-Hilbert problem and the method of nonlinear steepest descent is employed. This project is joint work with K. Schmidt (University of Central Florida) and R. Buckingham (University of Cincinnati).
March 9, 2026, 4:00–5:00 pm, zoom
Speaker: Alessandro Arsie (The University of Toledo)
Title: An introduction to Dirac geometry and reduction schemes for concurrent Dirac structures.
Abstract: I will provide first a gentle introduction to Dirac geometry, which is a way to unify pre-symplectic and Poisson geometry, as well as turning possibly singular Poisson structures into a perfectly smooth object.
After this, I will consider a particular situation of transferring Dirac structures which is the following: given an embedded submanifold X of a Dirac manifold (M, L_M) and given p: X -> Y a smooth surjective submersion, we want to derive the minimal set of conditions to transfer the Dirac structure L_M on M to a Dirac structure L_Y on Y. These conditions are however not compatible with concurrence, which is a generalization for Dirac structures of the notion of commuting Poisson pairs.
Then I will characterize a geometric structure, more precisely a vector bundle E\subset TM|_X that is a \emph{witness} for concurrence: it allows to transfer weakly concurrent Dirac structures on M to weakly concurrent Dirac structures on Y. We show that the Marsden-Ratiu reduction in Poisson geometry is exactly a special case of this construction. Furthermore, in the presence of a Hamiltonian action of a Lie group G on L_M, there is a natural candidate for a witness E.
The main results carry over to the case of complex Dirac structures. This allows us to give an extension of the bi-Hamiltonian reduction of Casati, Magri e Pedroni in terms of our framework and provide a (conjectural) interpretation of it in terms of complex Dirac structures.
This talk is based on a joint work with Dan Aguero, Pedro Frejlich and Igor Mencattini.
March 16, 2026, 4:00–5:00 pm, East Hall 1866
Speaker: Deniz Bilman (University of Cincinnati)
Title: Extreme Superposition: Rogue Waves of Infinite Order, Universality, and Anomalous Temporal Decay
Abstract: Focusing nonlinear Schrödinger equation serves as a universal model for the amplitude of a wave packet in a general one-dimensional weakly-nonlinear and strongly-dispersive setting that includes water waves and nonlinear optics as special cases. Rogue waves of infinite order are a novel family of solutions of the focusing nonlinear Schrödinger equation that emerge universally in a particular asymptotic regime involving a large-amplitude and near-field limit of a broad class of solutions of the same equation. In this talk, we will present several recent results on the emergence of these special solutions along with their interesting asymptotic and exact properties. Notably, these solutions exhibit anomalously slow temporal decay and are connected to the third Painlevé equation. Finally, we will extend the emergence of rogue waves of infinite order to the first several flows of the AKNS hierarchy—allowing for arbitrarily many simultaneous flows—and report on recent work regarding their space-time asymptotic behavior under a general flow from the hierarchy.
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
April 20, 2026, 4:00–5:00 pm, East Hall 1866
Speaker: Robert Buckhingham (University of Cincinnati)
Title: TBA
Abstract: TBA