“Mathematics Days in Sofia”

Section “Differential Equations and Mathematical Physics”

Participants

Invited Speakers

  • Fioralba Cakoni, Rutgers University, USA

  • John Schotland, Yale University, USA

  • Kiril Datchev, Purdue University, USA

  • Maciej Zworski, University of California, Berkeley, USA

  • Petar Topalov, Northeastern University, USA

Contributors

  • Agnieszka Bartłomiejczyk, Gdansk University of Technology, Poland

  • Atanas Stefanov, University of Alabama at Birmingham, USA

  • Georgi Boyadzhiev, Institute of Mathematics and Informatics, Bulgaria

  • Milena Stanislavova, University of Alabama at Birmingam, USA

  • Nikola Kamburov, Pontificia Universidad Catolica de Chile

  • Piotr Bartłomiejczyk, Gdańsk University of Technology, Poland

  • Vesselin Vatchev, University of Texas Rio Grande Valley, USA

Program and Abstracts

A perplexing question in scattering theory is whether there are incoming time harmonic waves, at particular frequencies, that are not scattered by a given inhomogeneity, in other words the inhomogeneity is invisible to probing by such waves. We refer to wave numbers, that correspond to frequencies for which there exists a non-scattering incoming wave, as non-scattering. This question is inherently related to the solution of inverse scattering problem for inhomogeneous media. The attempt to provide an answer to this question has led to the so-called transmission eigenvalue problem with the wave number as the eigen-parameter. This is non-selfadjoint eigenvalue problem with challenging mathematical structure. The non-scattering wave numbers form a subset of real transmission eigenvalues. A positive answer to the existence of non-scattering wave numbers was already known for spherical inhomogeneities and a negative answer was given for inhomogeneities with corners. Up to very recently little was known about non-scattering inhomogeneities that are neither spherical symmetric nor having support that contains a corner. In this presentation we discuss some new results for general inhomogeneities including anisotropic.
We present the state of the art on the spectral theory for the transmission eigenvalue problem. Then we examine necessary conditions for a transmission eigenvalue to be non-scattering wave number. These conditions are formulated in terms of the regularity of the boundary and refractive index of the inhomogeneity, using free boundary methods.
This presentation is based on joint works with Michael Vogelius and Jingni Xiao.

Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe recent work on a real-space formulation of quantum electrodynamics for single photons interacting with two-level atoms. It is shown that the probability amplitude of a photon obeys a nonlocal partial differential equation. Applications to quantum optics in random media will be described, where there is a close relation to kinetic equations for PDEs with random coefficients.

Two-dimensional scattering theory comes with special challenges due to singularities which appear at low frequencies. I will present a new technique, which is elementary and robust, for dealing with these singularities, focusing on the fundamental example of obstacle scattering, where the leading singularities are given in terms of the obstacle’s logarithmic capacity or Robin constant.  The technique is based on a resolvent identity of Vodev.

We expect similar results to hold for more general scattering problems, including singular magnetic vector potentials, corresponding to the Aharonov–Bohm effect in quantum mechanics. This talk is based on joint work with Tanya Christiansen, and joint work in progress with Tanya Christiansen and Mengxuan Yang.

The connections between the formation of internal waves in fluids, spectral theory, and homeomorphisms of the circle were investigated by oceanographers in the 90s and resulted in novel experimental observations (Leo Maas et al, 1997). The specific homeomorphism is given by a “chess billiard” and has been considered by many authors (Fritz John 1941, Vladimir Arnold 1957, Jim Ralston 1973…). The relation between the nonlinear dynamics of this homeomorphism and linearized internal waves provides a striking example of classical/quantum correspondence (in a classical and surprising setting of fluids!). I will illustrate the results with numerical and experimental examples and explain how classical concepts such as rotation numbers of homeomorphisms (introduced by Henri Poincare) are related to solutions of the Poincare evolution problem (so named by Elie Cartan). The talk is based on joint work with Semyon Dyatlov and Jian Wang. I will also mention recent progress by Zhenhao Li on the case of Diophantine rotation numbers.

We develop a framework for studying quasi-periodic maps and diffeomorphisms on Rn.
As an application, we prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on Rn. In particular, the equation preserves the spatial quasi-periodicity of the initial data. Several results on the analytic dependence of solutions on the time and the initial data will be discussed.
(This is a joint work with Xu Sun.)

Low-grade gliomas (LGGs) are primary brain tumours which evolve very slowly in time, however, inevitably cause patient death. In [1], we consider a PDE version of the previously proposed ODE model that describes the changes in the densities of functionally alive LGGs cells and cells that are irreversibly damaged by chemotherapy treatment. Besides the basic mathematical properties of the model, we study the possibility of the existence of travelling wave solutions in the framework of Fenichel invariant manifold theory. The minimum speeds of the travelling wave solutions were estimated. Obtained analytical results are illustrated by the numerical simulations.

This is joint work with Marek Bodnar, Magdalena U. Bogdanska and Monika J. Piotrowska.

Reference

[1] Bart lomiejczyk A., Bodnar M., Bogdanska M.U., Piotrowska M.J., Travelling waves for low-grade glioma growth and response to chemotherapy model, submitted

Abstract

In this talk is considered the validity of comparison and maximum principles for systems of second order PDEs with fully non-linear principal symbol. Interior and boundary maximum principles, as well as comparison principle, are given for quasi – monotone systems of degenerate elliptic and parabolic equations. Applications as existence theorems are given as well.
In this talk, we showcase the advantages of using uniform resolvent estimates close to the imaginary axis to investigate the asymptotic behavior of the solution semigroups in the case of spectrally stable generators. First, we consider the spectrally stable periodic steady states of the Lugiato-Lefever model of optical fibers and show that the solution semigroup obeys optimal exponential decay estimates. Moreover, these resolvent estimates can be used to prove that the solutions are asymptotically stable with phase. Our second example is motivated by the linearization about standing waves of the NLS on a periodic domain. In this situation, we prove that the semigroup grows at most polynomially in time and is uniformly bounded on L2 when all eigenvalues of the generator have the same algebraic and geometric multiplicities.

Abstract

Map-based neuron models are important tools in modelling neural dynamics and sometimes can be considered as an alternative to usually computationally more expensive models based on continuous or hybrid dynamical systems. We study two discrete models of neuronal dynamics (see [1, 2]). The first model was introduced by Chialvo in 1995 and the second one by Courbage, Nekorkin and Vdovin in 2007. We show that their reduced one-dimensional versions can be treated as independent simple models of neural activity, where the input and the fixed value of the recovery variable are parameters. It occurs that these one-dimensional systems still display very rich and varied dynamics. We provide a detailed analysis of both periodic and chaotic behaviour of the models.

This is joint work with Frank Llovera Trujillo and Justyna Signerska-Rynkowska.

References

[1] Frank Llovera Trujillo, Justyna Signerska-Rynkowska, Piotr Bartłomiejczyk, Periodic and chaotic dynamics in a map-based neuron model, Mathematical Methods in the Applied Sciences 2023, 1-26, doi:10.1002/mma.mma.9118.

[2] Piotr Bartłomiejczyk, Frank Llovera Trujillo, Justyna Signerska-Rynkowska, Spike patterns and chaos in a map-based neuron model, submitted

Abstract

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