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The dynamics of hexagon patterns in rotating systems are investigated within the framework of modified Swift-Hohenberg equations that can be considered as simple models for rotating convection with broken up-down symmetry, e.g.…
We construct simple Lagrangians of vector fields which involve second derivatives, but nevertheless lead to second order field equations. These vector fields are, therefore, analogs of generalized Galileons. Our construction is given first…
A parameter dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non-self-adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet…
We derive a necessary and sufficient condition for Turing instabilities to occur in two-component systems of reaction-diffusion equations with Neumann boundary conditions. We apply this condition to reaction-diffusion systems built from…
For an activator-inhibitor reaction-diffusion system in a bounded three-dimensional domain $\Omega$ of $O(1)$ volume and small activator diffusivity of $O(\varepsilon^2)$, we employ a hybrid asymptotic-numerical method to investigate two…
Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is studied by means of the field theoretic renormalization group and the operator product expansion. The…
The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields…
We consider the three-dimensional Heisenberg group, equipped with any left-invariant metric, either Lorentzian or Riemannian. We completely classify their affine vector fields and investigate their relationship with Killing vector fields…
This paper concerns the behavior of the eigenfunctions and eigenvalues of the round sphere's Laplacian acting on the space of sections of a real line bundle which is defined on the complement of an even numbers of points in $S^2$. Of…
Scalar, vector and tensor harmonics on the three-sphere were introduced originally to facilitate the study of various problems in gravitational physics. These harmonics are defined as eigenfunctions of the covariant Laplace operator which…
In dimensions d= 1, 2, 3 the Laplacian can be perturbed by a point potential. In higher dimensions the Laplacian with a point potential cannot be defined as a self-adjoint operator. However, for any dimension there exists a natural family…
We address some global solvability issues for classes of smooth nonsingular vector fields $L$ in the plane related to cohomological equations $Lu=f$ in geometry and dynamical systems. The first main result is that $L$ is not surjective in…
Pseudo-Hermitian (including $\mathcal{PT}$-symmetric) field theories support phenomenology that cannot be replicated in standard Hermitian theories. We describe a concrete example in which the vortex solutions that are realised in a…
Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with boundary. In this article, we study the effects of the presence of a nontrivial conformal vector field on $(M^n,g)$. We used the wekk-known de-Rham Laplace…
A method of obtaining vector constants of motion for time-independent as well as time-dependent central fields is discussed. Some well-established results are rederived in this alternative way and new ones obtained.
Hermitian Hamiltonians with time-periodic coefficients can be analyzed via Floquet theory, and have been extensively used for engineering Floquet Hamiltonians in standard quantum simulators. Generalized to non-Hermitian Hamiltonians,…
The vector field problem is an important and classical problem in differential topology. In this survey we shall consider the vector field problem focusing mainly on the class of compact homogeneous spaces.
We begin with a review of the statements of non-linear, linear and mode stability of autonomous dynamical systems in classical mechanics, using symplectic geometry. We then discuss what the phase space and the Hamiltonian of general…
We examine the existence and stability of nonlinear discrete vortex solitons in a square lattice when the standard discrete Laplacian is replaced by a fractional version. This creates a new, effective site-energy term, and a coupling among…
In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems…