Project Leader: Prof. Jarosław Mederski
The research project focuses on nonlinear partial differential equations (PDEs) arising in electromagnetism, nonlinear optics, and materials science. We study problems involving the Schrödinger operator, the curl operator, as well as other elliptic and evolutionary PDEs. The main subject is the analysis of Maxwell’s equations with nonlinear polarization, which describe phenomena of central importance in physics and engineering. Nonlinear media, such as Kerr-type materials, play a crucial role in nanotechnology, enabling the design of structures smaller than the wavelength of light. These materials exhibit new optical properties that cannot be captured by linear models.
The aim of the project is to investigate time-harmonic electromagnetic waves in nonlinear media using analytical methods. These techniques are relatively new in the context of nonlinear curl–curl equations, and we expect to develop novel mathematical tools. We focus on the existence of ground and bound state solutions for semilinear equations with various elliptic operators. This allows us to model light propagation in metamaterials whose response depends on field intensity. Nonlinear Maxwell equations are also highly relevant in the description of materials such as graphene and topological insulators exposed to strong electromagnetic fields. Another key goal is the rigorous study of nonlocal models, such as those involving the fractional Laplacian (fractional operators) or Waldenfels operators, which more accurately reflect real-world phenomena than classical local models. Such models are widely investigated in physics, biology, and economics. We also address nonclassical problems with singular data, for example in nonsmooth domains, with measure data, or under weak regularity assumptions.
Finally, we examine the time evolution of advanced materials, taking into account deformation, diffusion, chemical reactions, and the dynamic behavior of biomaterials. Mathematical modeling plays a central role here—it enables the analysis of structural stability, adaptive processes, self-organization, and the existence of trajectories satisfying complex boundary conditions. To effectively describe these phenomena, we will employ modern dimensional reduction methods, combining advanced analysis of Maxwell’s equations with simplified numerical models. We will also explore regularization techniques, such as lasso, originating from machine learning and statistics. These methods help reduce model complexity, prevent overfitting, and extract the most significant physical dependencies, which is particularly important in analyzing complex material systems.
The research tasks lie at the intersection of variational and topological methods, differential equations, functional analysis, harmonic analysis, statistics, machine learning, and mathematical physics. Potential collaborators in the project have diverse mathematical expertise—they work on nonlinear equations but often apply different approaches and tools. The project creates an opportunity to bring together researchers who have thus far worked independently. The potential applications of the expected results include a better understanding of physical models in nonlinear optics, nonlinear materials, and biomaterials, as well as the development of more efficient numerical methods for nonlinear PDEs. The anticipated outcome of the project is the proof of existence results for the studied problems and the development of new mathematical methods, which may also find applications in other areas of mathematical physics.