Related papers: Local Geometric Invariants of Integrable Evolution…
Methods to solve the relativistic hydrodynamic equations are a key computational kernel in a large number of astrophysics simulations and are crucial to understanding the electromagnetic signals that originate from the merger of…
We develop a covariant formalism to study nonlinear perturbations of dissipative and interacting relativistic fluids. We derive nonlinear evolution equations for various covectors defined as linear combinations of the spatial gradients of…
We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces…
A one-parameter generalization of the hierarchy of negative flows is introduced for integrable hierarchies of evolution equations, which yields a wider (new) class of non-evolutionary integrable nonlinear wave equations. As main results,…
We provide a concise introduction to the symmetry approach to integrability. Some results on integrable evolution and systems of evolution equations are reviewed. Quasi-local recursion and Hamiltonian operators are discussed. We further…
Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic…
The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in…
In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schr\"odinger equation, the complex…
The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the…
Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an…
Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina et al., J. Comput. Phys.,…
A vector calculus approach for the determination of advected invariants is presented for inviscid fluid flow. This approach describes invariants by means of Lie dragging of scalars, vectors, and skew-tensors with respect to the fluid…
The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat [3], is characterized by the property that to each solution of a differential equation, within the class, there corresponds a…
The development of the theory of three-dimensional harmonic mappings is considered. The new classes of mappings that generate three-dimensional harmonic functions are introduced. The physical interpretation of these mappings is applied to…
The link between 3D spaces with (in general, non-constant) curvature and quantum deformations is presented. It is shown how the non-standard deformation of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that…
Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a…
The cyclic evolutions and associated geometric phases induced by time-independent Hamiltonians are studied for the case when the evolution operator becomes the identity (those processes are called {\it evolution loops}). We make a detailed…
We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…
In this paper, we consider time-varying real analytic vector fields as curves on the space of real analytic vector fields. Using a suitable topology on the space of real analytic vector fields, we study and characterize different properties…
We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of {\it capacitary measures} with respect to the…