Related papers: The Degasperis-Procesi equation as a non-metric Eu…
An open problem about finite geometric progressions in syndetic sets leads to a family of diophantine equations related to the commutativity of translation and multiplication by squares.
We study the geodesic flow of a compact surface without conjugate points and genus greater than one and continuous Green bundles. Identifying each strip of bi-asymptotic geodesics induces an equivalence relation on the unit tangent bundle.…
In this article we investigate the dynamical properties of the geodesic flow for a proper metric space endowed with a proper action by isometries of a group with a contracting element. We show that the existence of a contracting isometry is…
We use the r-matrix formulation to show the integrability of geodesic flow on an $N$-dimensional space with coordinates $q_k$, with $k=1,...,N$, equipped with the co-metric $g^{ij}=e^{-|q_i-q_j|}\big(2-e^{-|q_i-q_j|}\big)$. This flow is…
The motion of compressible, inviscid fluid under the constant pressure on a rotating sphere is studied. The hodograph equations for the corresponding Euler equation are presented. They provide us with the class of solutions of the Euler…
This paper concerns the classical dynamics of three coupled rotors: equal masses moving on a circle subject to attractive cosine inter-particle potentials. It is a simpler variant of the gravitational three-body problem and also arises as…
In this paper, the steady inviscid flows with radial symmetry for the isothermal Euler system are studied in an annulus. We present a complete classification of transonic radially symmetric flow patterns in term of physical boundary…
The space of quadrilaterals with fixed side lengths is an elliptic curve. Darboux used this to prove a porism on foldings. In this article, the space of oriented quadrilaterals is studied on the base of biquadratic equations between their…
The scientific question resolved by this paper is that the continuity equation appears as an equivalent language of the system of first-order linear ODE. The main result characterizes the fact that the continuity equation contains…
We develop an abstract theory of flows of geometric $H$-structures, i.e., flows of tensor fields defining $H$-reductions of the frame bundle, for a closed and connected subgroup $H\subset SO(n)$, on any connected and oriented $n$-manifold…
The Degasperis-Procesi equation with self-consistent sources(DPESCS) is derived. The Lax representation and the conservation laws for DPESCS are constructed. The peakon solution of DPESCS is obtained.
On a smooth, compact and oriented manifold without boundary, we give a complete description of the correlation function of a Morse-Smale gradient flow satisfying a certain nonresonance assumption. This is done by analyzing precisely the…
The article provides an analytical solution of the Navier-Stokes equations for the case of the steady flow of an incompressible fluid between two uniformly co-rotating disks. The solution is derived from the asymptotical evolution of…
A fourth-order dispersive flow equation for closed curves on the canonical two-dimensional unit sphere arises in some contexts in physics and fluid mechanics. In this paper, a geometric generalization of the sphere-valued model is…
Consider inviscid fluids in a channel {-1<y<1}. For the Couette flow v_0=(y,0), the vertical velocity of solutions to the linearized Euler equation at v_0 decays in time. At the nonlinear level, such inviscid damping has not been proved.…
The paper introduces a general framework for derivation of continuum equations governing meso-scale dynamics of large particle systems. The balance equations for spatial averages such as density, linear momentum, and energy were previously…
We study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real…
Hodograph equations for the Euler equation in curved spaces with constant pressure are discussed. It is shown that the use of known results concerning geodesics and associated integrals allows to construct several types of hodograph…
We investigate the emergence of finite-amplitude non-zonal flows on the sphere $\mathbb{S}^2$ arising from stationary solutions to the 2D Euler equations. By restricting the Laplace-Beltrami eigenspace to the invariant subspace of the…
The parametric nonlinear Schrodinger equation models a variety of parametrically forced and damped dispersive waves. For the defocusing regime, we derive a normal velocity for the evolution of curved dark-soliton fronts that represent a…