Related papers: Dynamical systems on infinitely sheeted Riemann su…
We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation…
Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links…
Perfect fluid spacetimes admitting a kinematic self-similarity of infinite type are investigated. In the case of plane, spherically or hyperbolically symmetric space-times the field equations reduce to a system of autonomous ordinary…
We develop, and implement in a Finite Volume environment, a density-based approach for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations…
We introduce a geometric dynamical system where iteration is defined as a cycling composition of different maps acting on a space composed of three or more lines in $\mathbb{R}^2$. This system is motivated by the dynamics of iterated…
We show that there exist closed three-dimensional Riemannian manifolds where the incompressible Euler equations exhibit smooth steady solutions that are isolated in the $C^1$-topology. The proof of this fact combines ideas from dynamical…
It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian…
This report addresses the solution of Riemann problems for hyperbolic equations when the nonlinear characteristic fields loose their genuine nonlinearity. In this context, exact solvers for nonconvex 1D Riemann problems are developed. First…
The topology of complex classical paths is investigated to discuss quantum tunnelling splittings in one-dimensional systems. Here the Hamiltonian is assumed to be given as polynomial functions, so the fundamental group for the Riemann…
Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with inverse power law form for power spectra of temporal fluctuations on all space-time…
This work presents a new finite volume framework for solid dynamics based on a momentum-deformation formulation. Building on the C-TOUCH methodology [1], a novel Roe-type Riemann solver is developed to enhance the stability and accuracy of…
We outline a 2D algorithm for solving incompressible flow--structure interaction problems for mixed rigid/soft body representations, within a consistent framework based on the remeshed vortex method. We adopt the one--continuum formulation…
The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks,…
We study the complexification of the one-dimensional Newtonian particle in a monomial potential. We discuss two classes of motions on the associated Riemann surface: the rectilinear and the cyclic motions, corresponding to two different…
The dynamical equations of clarinet-like systems are known to be reducible to a non-linear iterated map within reasonable approximations. This leads to time oscillations that are represented by square signals, analogous to the Raman regime…
We study laminar thin film flows with large distortions in the free surface using the method of averaging across the flow. Two concrete problems are studied: the circular hydraulic jump and the flow down an inclined plane. For the circular…
We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems involving systems of curves and the jump problem. We obtain an explicit expression for the scattering matrix in terms of…
The capability to accurately predict flood flows via numerical simulations is a key component of contemporary flood risk management practice. However, modern flood models lack the capacity to accurately model flow interactions with linear…
In this paper, we study the long time dynamics of small solutions to Schr\"odinger map flows from $\Bbb R$ to Riemannian surfaces. The results are threefold. (i) We prove that for general Riemannian surface targets the points with some…
The Riemann problem for first-order hyperbolic systems of partial differential equations is of fundamental importance for both theoretical and numerical purposes. Many approximate solvers have been developed for such systems; exact solution…