Special Session 23: Evolution Equations and Integrable Systems

Scattering for radial solutions of focusing supercritical wave equations

Guher Camliyurt Virginia Tech USA Co-Author(s):    Carlos Kenig

Abstract: This talk mainly focuses on the scattering of radial bounded solutions to focusing semi-linear wave equations with energy supercritical nonlinearity in four dimensions. We first review earlier applications of the concentration compactness and rigidity methods in energy-critical and energy-supercritical settings. As most of these results address the problem in three dimensions, we explore the unique challenges posed by analogous problems in even dimensions. We then investigate how rigidity arguments, originally developed for three dimensions and successfully generalized to higher odd dimensions, can be adapted to address these new difficulties in even-dimensional settings.

Linear Evolution Equations Revisited via the Fokas Unified Transform Method

Andreas Chatziafratis National and Kapodistrian University of Athens Greece Co-Author(s):

Abstract: We survey some of our recent research results relating to construction of explicit solutions (closed-form integral representations), in the classical sense, as well as to qualitative theory for non-homogeneous initial-boundary-value and interface problems for a variety of linear (systems of) evolution partial differential equations (PDE) with constant and with variable coefficients. Such PDE emerge in connection to a plethora of natural phenomena and mathematical models in physics, biology, chemical engineering, finance and other applied sciences; examples include continuum mechanics, heat transfer, biomedicine, electron optics, and battery technology. Our research program relies on, explores and extends the applicability of the celebrated complex-analytic unified transform method of Fokas. This is joint work with a large global network of collaborators. Notable findings include: long-range instabilities (see e.g. [A. Chatziafratis, T. Ozawa, S.-F. Tian, “Rigorous analysis of the unified transform method and long-range instabilities for the inhomogeneous time-dependent Schrödinger equation on the quarter-plane”, Math. Annalen, 2023] and [A. Chatziafratis, L. Grafakos, S. Kamvissis, Long-range instabilities for linear evolution PDE on semi-bounded domains via the Fokas method, Dyn. PDE, 2024]), time-asymptotic break-down effects (e.g. [J.L. Bona, A. Chatziafratis, H. Chen, S. Kamvissis, “The linear BBM-equation on the half-line revisited”, Lett. Math. Phys., 2024]), and counter-examples to solution uniqueness (e.g. [A. Chatziafratis, A. Miranville, G. Karali, A.S. Fokas, E.C. Aifantis, "Higher-order diffusion and Cahn-Hilliard-type models revisited on the half-line", Math. Models Methods Appl. Sci., 2025] and [A. Chatziafratis, S. Kamvissis, “Infinity of solutions to initial-boundary value problems for linear constant-coefficient evolution PDEs on semi-infinite intervals”, Bull. London Math. Soc., 2025])

The nonlinear Schr\odinger equation on the half-space

Alex Himonas University of Notre Dame USA Co-Author(s):    Fangchi Yan

Abstract: This work studies the initial-boundary value problem for both the linear Schr\odinger equation and the cubic nonlinear Schr\odinger equation on the half-space in higher dimensions ($n\ge 2$). First, the forced linear problem is solved on the half-space via the Fokas method and then using the obtained solution formula new and interesting linear estimates are derived with data and forcing in appropriate spaces. Second, the well-posedness of the nonlinear problem on the half-space is proved with initial data in Sobolev spaces $H^s(\mathbb{R}^n_+)$, with $s>\frac{n}{2}-1$, and boundary data in natural Bourgain spaces $\mathcal{B}^s$ that reflect the boundary regularity of the linear problem. The proof method consists of showing that the iteration map defined via the Fokas solution formula is a contraction by establishing sharper trilinear estimates. The presence of the boundary introduces solution spaces that involve temporal Bourgain spaces.

A Predictive Framework for Glioblastoma Progression

Curtis A Holliman The Catholic University of America USA Co-Author(s):    Francesca Cozzi

Abstract: Glioblastoma remains one of the most challenging malignancies to treat, owing in part to its aggressive and heterogeneous progression. Developing reliable predictive frameworks for tumor dynamics has the potential to inform clinical decision-making and improve patient outcomes. In this talk, we present a computational approach to modeling glioblastoma progression, integrating mathematical and data-driven techniques to capture key features of tumor behavior. We discuss the theoretical foundations of our framework, its predictive capabilities, and directions for future development.

On Special Properties of Solutions to the Benjamin-Bona-Mahony Equation

Christian Hong University of California Santa Barbara USA Co-Author(s):    Gustavo Ponce

Abstract: This work is concerned with the Benjamin-Bona-Mahony equation. This model was deduced as an approximation to the Korteweg-de Vries equation in the description of the unidirectional propagation of long waves. Our goal is to present some results on unique continuation and regularity properties of solutions to the associated initial value problem and intial periodic boundary value problems.

Hamiltonian Transformation Theory and Dispersive PDEs

Adilbek Kairzhan Nazarbayev University Kazakhstan Co-Author(s):    Philippe Guyenne, Catherine Sulem

Abstract: In this talk, we present the derivation of a Hamiltonian Dysthe equation for the slowly varying envelope of modulated wavetrains based on a Hamiltonian formulation of the water wave problem and by applying techniques from Hamiltonian theory. The models we consider include 2d and 3d surface gravity waves and 2d water waves with constant vorticity. Our method provides a procedure to reconstruct the surface elevation from the wave envelope, based on the Birkhoff normal form transformation to eliminate all non-resonant triads. The talk is based on a series of works with Catherine Sulem (University of Toronto) and Philippe Guyenne (University of Delaware).

Integral solutions of evolution equations

Konstantinos Kalimeris Academy of Athens Greece Co-Author(s):

Abstract: In this talk we consider initial-boundary value problems (IBVPs) for linear evolution partial differential equations (PDEs) with time-dependent coefficients. We present (for the first time) integral representations of solutions for a big family of these problems; the derivation of these solutions is based on extensions of the unified transform (UT). Based on these solutions we discuss the well-posedness of these IBVPs for certain linear and non-linear PDEs. Finally, we show that, for certain evolution partial PDEs, the solution of IBVPs on a finite interval can be reconstructed as a superposition of solutions to two associated IBVPs posed on the half-line, which provides a completely new perspective on studying this large class of problems.

Stokes Phenomenon in CR geometry

Ilya Kossovskiy Southern University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: In our joint work with Shafikov (JEMS, 2016), we proved that the formal versus holomorphic equivalence problems in CR geometry are distinct. This shows, in particular, that CR geometry enjoys the Stokes Phenomenon: the existence of additional holomorphic invariants supplementing a formal normal form. Describing the Stokes phenomenon explicitly has been an open problem since then. In our joint work with Laurent Stolovitch, we make a significant progress in solving this problem. In particular, we show that the respective (formal versus holomorphic) moduli space of analytic CR hypersurfaces in C^2 is, somewhat surprisingly, finite-dimensional (and generically empty). The construction that we use is based on the Multisummability theory in Dynamical Systems.

The complex Ginzburg-Landau equation on a finite interval and chaos suppression via a finite-dimensional boundary feedback stabilizer. Part I: Well-posedness

Dionyssis Mantzavinos University of Kansas USA Co-Author(s):    Turker Ozsari, Kemal Cem Yilmaz

Abstract: This is the first of two talks on the well-posedness and boundary stabilization of the initial-boundary value problem for the complex Ginzburg-Landau equation on a finite interval. This first talk focuses on the well-posedness of the open loop model. More precisely, it establishes a local well-posedness theory for the open loop model in $L^2$-based fractional Sobolev spaces in the case of Dirichlet-Neumann type inhomogeneous mixed boundary conditions. This local well-posedness result is based on linear estimates derived by using the weak solution formula obtained via the unified transform (also known as the Fokas method). The global well-posedness properties of the open loop model in the presence of inhomogeneous boundary conditions are also discussed. In a sequel talk by Turker Ozsari, the results discussed here for the open loop model will be employed for the analysis of the rapid boundary feedback stabilization problem and the design of a nonlocal controller via a finite number of Fourier modes of the state of solution.

The complex Ginzburg--Landau equation on a finite interval and chaos suppression via a finite-dimensional boundary feedback stabilizer. Part II: Control

Turker Ozsari Bilkent University Turkey Co-Author(s):    Dionyssios Mantzavinos; Kemal Cem Yilmaz

Abstract: This is the second of two talks on well-posedness and boundary stabilization for the complex Ginzburg--Landau (CGL) equation on a finite interval. In the prequel talk (D. Matzavinos, Part I), the open-loop Dirichlet--Neumann initial-boundary value problem is shown to be locally (and, in suitable regimes, globally) well posed at low regularity via sharp spatiotemporal estimates obtained from a unified-transform (Fokas method) representation formula. Building on those estimates, I present a rapid boundary feedback stabilization (chaos suppression) scheme using Neumann actuation: a nonlocal controller depending only on finitely many Fourier modes, based on a slow--fast decomposition into a finite-dimensional ``slow'' part and a rapidly decaying tail. We quantify (i) how many modes guarantee exponential stabilization at a prescribed decay rate, and (ii) the minimal number ensuring stabilization at some exponential rate. Closed-loop well-posedness follows locally by combining Part I spatiotemporal bounds with the feedback structure, while global energy solutions are obtained from the stabilization estimates. Uniqueness is derived by reducing to a homogeneous-boundary problem through a bounded, invertible Volterra-type integral transform on Sobolev spaces. Numerical simulations illustrating chaos suppression will also be discussed. This research was supported by TUBITAK 1001 Grant 122F084.

Regularity of solution of periodic evolution equations

Beatrice Pelloni Heriot-Watt University Scotland Co-Author(s):    Lyonell Boulton

Abstract: I will discuss the regularity of solution of periodic evolution equations starting from an initial condition of low regularity. Specifically, I will present many more general instances of what was known as the Talbot effect, that is felt as an echo of the linear effect also in the solution on nonlinear problems.

Waves in cosmological background with static Schwarzschild radius in the expanding universe

Karen Yagdjian University of Texas Rio Grande Valley USA Co-Author(s):

Abstract: The latest astrophysical observational data confirm that the universe is expanding with acceleration and that the masses of the black holes are changing over time. This raises the question of the solutions to partial differential equations that model the propagation of waves in a cosmological background with black holes in an expanding universe. We prove the existence of global in time small data solutions of semilinear Klein-Gordon equations in space-time with a black hole with time-dependent mass and static Schwarzschild radius in the expanding universe. We indicate that the damping term of the covariant Klein-Gordon equation is independent of location in the background only if the mass of the black hole is proportional to the scale factor of expansion. This is where the cosmological principle comes into play.

A higher order cubic NLS equation on the half-line

Fangchi Yan Virginia Tech USA Co-Author(s):    Alex Himonas

Abstract: This work studies the initial-boundary value problem on the half-line for the cubic nonlinear Schr\odinger equation with a dispersion of order $m=2, 4, \cdots$. The main result obtained is the optimal well-posedness of this problem when the initial data belong in the spatial Sobolev spaces $H^s(0,\infty)$, $s>-\frac14m +\frac12$, and the boundary data belong in appropriate temporal Sobolev spaces suggested by the time regularity of the linear problem. The proof is based on the Fokas solution formula for the forced linear problem and the linear estimates obtained for this solution in Bourgain spaces. Deriving sharp trilinear estimates suggested by the linear estimates and then applying them, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space.