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In classical General Relativity, the way to exhibit the equations for the gravitational waves is based on two "tricks" allowing to transform the Einstein equations after linearizing them over the Minkowski metric. With specific notations…
Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is…
The equations for gravitational plane waves produced by a typical binary system as a solution of linear approximation of Einstein equations is derived. The dynamics of the corresponding gravitational field is analyzed in a 4-dimensional…
A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. For a probability distribution, the length along a path between two points can be defined as the amount of probability mass accumulated…
The present study describes, first, an efficient algorithm for computing capillary-gravity solitary waves solutions of the irrotational Euler equations with a free surface and, second, provides numerical evidences of the existence of an…
Within the closed universe, we obtain the amplitude and frequency of gravitational waves in the terms of discrete wave numbers, wave propagation time, and cosmological constant using the deviation equation in the first-order perturbed…
We study Weil-Petersson (WP) geodesics with narrow end invariant and develop techniques to control length-functions and twist parameters along them and prescribe their itinerary in the moduli space of Riemann surfaces. This class of…
Wave turbulence is by nature a multiple time scale problem for which there is a natural asymptotic closure. The main result of this analytical theory is the kinetic equation that describes the long-time statistical behaviour of such…
Geodesic problems involve computing trajectories between prescribed initial and final states to minimize a user-defined measure of distance, cost, or energy. They arise throughout physics and engineering -- for instance, in determining…
Numerical simulations of the unidirectional random waves are performed within the Korteweg -de Vries equation to investigate the statistical properties of surface gravity waves in shallow water. Nonlinear evolution shows the relaxation of…
According to Schroedinger's ideas, classical dynamics of point particles should correspond to the " geometrical optics " limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using…
Starting with an exact and simple geodesic, we generate approximate geodesics by summing up higher-order geodesic deviations within a General Relativistic setting, without using Newtonian and post-Newtonian approximations. We apply this…
We demonstrate that geodesics in exact vacuum Kundt gravitational waves may exhibit a highly complicated behaviour. In fact, as in the previously studied case of non-homogeneous pp-waves, for specific choices of the structural function the…
The celebrated geometric control condition of Bardos, Lebeau, and Rauch is necessary and sufficient for wave observability [1,7] and exact controllability. It requires that any point in phase-space be transported by the generalized geodesic…
We study the free motion of a massive particle moving in the background of a Finslerian deformation of a plane gravitational wave in Einstein's General Relativity. The deformation is a curved version of a one-parameter family of…
We survey recent developments on the analysis of Gauss--Codazzi--Ricci equations, the first-order PDE system arising from the classical problem of isometric immersions in differential geometry, especially in the regime of low Sobolev…
We consider the geodesic deviation equation, describing the relative accelerations of nearby particles, and the Raychaudhuri equation, giving the evolution of the kinematical quantities associated with deformations (expansion, shear and…
We solve explicitly the geodesic equation for a wide class of (pseudo)-Riemannian homogeneous manifolds (G/H,m), including those with G compact, as well as non-compact semisimple Lie groups, under a simple algebraic condition for the metric…
An axiomatic theory of classical nondissipative waves is proposed that is constructed based on the definition of a wave as a multidimensional oscillator. Waves are represented as abstract vectors $|\psi\rangle$ in the appropriately defined…
We investigate the propagation of locally plane, small-amplitude, monochromatic gravitational waves through cold compressible interstellar gas in order to provide a more accurate picture of expected waveforms for direct detection. The…