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The transition between kinetic and hydrodynamic regimes of the one-dimensional two-stream instability is numerically analyzed, and the correction coefficients to the well-known textbook formulae are calculated. The approximate expressions…
We study $(2,2)$ and $(4,4)$ supersymmetric theories with superspace higher derivatives in two dimensions. A characteristic feature of these models is that they have several different vacua, some of which break supersymmetry. Depending on…
A general Hamiltonian theory for the adiabatic motion of relativistic charged particles confined by slowly-varying background electromagnetic fields is presented based on a unified Lie-transform perturbation analysis in extended phase space…
In this paper we review recent progress on relativistic hydrodynamics in (2 + 1)-dimensions with an external magnetic field. We discuss the formalism allowing for momentum loss due to the explicit and spontaneous breaking of translation…
We address the fundamental difference between solid-solid and liquid-liquid phase transitions within the Ericksen's nonlinear elasticity paradigm. To highlight ideas, we consider the simplest nontrivial 2D problem and work with a…
This paper deals with the numerical (finite volume) approximation of reaction-diffusion systems with relaxation, among which the hyperbolic extension of the Allen--Cahn equation represents a notable prototype. Appropriate discretizations…
We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties…
Dimensionality serves as an indispensable ingredient in any attempt to formulate the low-dimensional physics, and studying the dimensional crossover at a fundamental level is challenging. The purpose of this work is to study the…
Relativistic elasticity on an arbitrary spacetime is formulated as a Lagrangian field theory which is covariant under spacetime diffeomorphisms. This theory is the relativistic version of classical elasticity in the hyperelastic, materially…
We attempt to give a bird's eye view of the physical mechanisms leading to anomalous relaxation, and the relation of this phenomenon with anomalous diffusion and transport. Whereas in some cases these two notions are indeed deeply related,…
We consider relativistic hydrodynamics in the limit where the number of spatial dimensions is very large. We show that under certain restrictions, the resulting equations of motion simplify significantly. Holographic theories in a large…
This paper is based on a series of talks given at the ESI program on 'Mathematical Perspectives of Gravitation Beyond the Vacuum Regime' in February 2022. It is meant to be an introduction to the field of relativistic elasticity for readers…
In applied mathematics generally and fluid dynamics in particular, the role of complex variable methods is normally confined to two-dimensional motion and the association of points with complex numbers via the assignment w = x+i y. In this…
Understanding the relaxation of a system towards equilibrium is a longstanding problem in statistical mechanics. Here we address the role of long-range interactions in this process by considering a class of two-dimensional or geophysical…
The three-dimensional elastic-plastic deformation is considered. The catastrophe theory underlies the construction of this process model. It was shown that the variety of stable states consists on elastic states and can be depicted as a…
We derive and analyze a novel approach for modeling and computing the mechanical relaxation of incommensurate 2D heterostructures. Our approach parametrizes the relaxation pattern by the compact local configuration space rather than real…
This work explores the solvability of a sixth-order Cahn--Hilliard equation with an inertial term, which serves as a relaxation of a higher-order variant of the classical Cahn--Hilliard equation. The equation includes a source term that…
This document is an introduction to and review of two-dimensional mathematical physics. The reader is introduced to the subject matter primarily through problems, which are presented along with detailed worked solutions. For each chapter,…
The effects of three-dimensional perturbations in two-dimensional turbulence are investigated, through a conformal field theory approach. We compute scaling exponents for the energy spectra of enstrophy and energy cascades, in a strong…