On the existence of traveling waves in the 3D Boussinesq system

We extend earlier work on traveling waves in premixed flames in a gravitationally stratified medium, subject to the Boussinesq approximation. For three-dimensional channels not aligned with the gravity direction and under the Dirichlet boundary conditions in the fluid velocity, it is shown that a non-planar traveling wave, corresponding to a non-zero reaction, exists, under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the Prandtl and Rayleigh numbers, or when the advection term in the flow equations is neglected.Comment: 15 pages, to appear in Communications in Mathematical Physic

Mathematical Theory of Water Waves

The water-wave problem is the study of the two- and three-dimensional flow of a perfect fluid bounded above by a free surface subject to the forces of gravity and surface tension. From a mathematical viewpoint, the water-wave equations pose surprisingly deep and subtle challenges for mathematical analysis. The governing equations are widely accepted and there has been substantial research into their validity and limitations. However, a rigorous theory of their solutions is extremely complex due not only to the fact that the water-wave problem is a classical free-boundary problem (where the problem domain, specifically the water surface, is one of the unknowns), but also because the boundary conditions (and, in some cases, the equations) are strongly nonlinear. In contrast to other meetings on water waves, which usually focus upon modelling and numerical issues, this workshop was devoted to the rigorous mathematical theory for the exact hydrodynamic equations

Complex inflaton field in quantum cosmology

We investigate the cosmological model with the complex scalar self-interacting inflaton field non-minimally coupled to gravity. The different geometries of the Euclidean classically forbidden regions are represented. The instanton solutions of the corresponding Euclidean equations of motion are found by numerical calculations supplemented by the qualitative analysis of Lorentzian and Euclidean trajectories. The applications of these solutions to the no-boundary and tunneling proposals for the wave function of the Universe are studied. Possible interpretation of obtained results and their connection with inflationary cosmology is discussed. The restrictions on the possible values of the new quasi-fundamental constant of the theory-non-zero classical charge-- are obtained. The equations of motion for the generalized cosmological model with complex scalar field are written down and investigated. The conditions of the existence of instanton solutions corresponding to permanent values of an absolute value of scalar field are obtained.Comment: 34 pages with 2 gif figures, mprocl.sty, To appear in International Journal of Modern Physics

Gauge Fields, Fermions and Mass Gaps in 6D Brane Worlds

We study fluctuations about axisymmetric warped brane solutions in 6D minimal gauged supergravity. Much of our analysis is general and could be applied to other scenarios. We focus on bulk sectors that could give rise to Standard Model gauge fields and charged matter. We reduce the dynamics to Schroedinger type equations plus physical boundary conditions, and obtain exact solutions for the Kaluza-Klein wave functions and discrete mass spectra. The power-law warping, as opposed to exponential in 5D, means that zero mode wave functions can be peaked on negative tension branes, but only at the price of localizing the whole Kaluza-Klein tower there. However, remarkably, the codimension two defects allow the Kaluza-Klein mass gap to remain finite even in the infinite volume limit. In principle, not only gravity, but Standard Model fields could `feel' the extent of large extra dimensions, and still be described by an effective 4D theory.Comment: 33 pages, 2 figures; typesetting problem fixed ({\o}replaced by \omega

Numerical solution of plane dynamic problems of the stiffened cylindrical shells elliptical cross – section theory under non-stationary load

Розглянуто плоску задачу про вимушені коливання поздовжньо підкріплених циліндричних оболонок еліптичного перерізу під дією розподіленого нестаціонарного навантаження. Наведено постановку та розроблено чисельний алгоритм розв’язування задачі. Наведено приклади розрахунку плоских задач динамічної поведінки підкріплених оболонок та проведено аналіз отриманих чисельних результатів.In this paper, in the framework of the theory of shells and rods of Timoshenko-type, the problem of forced vibrations of longitudinal – cross stiffened cylindrical shells of elliptical cross section is considered. Numerical algorithm for solving the problem is constructed. The results of analysis of nonstationary vibrations of longitudinal - cross stiffened shells are presented. To derive the vibration equations for stiffened cylindrical shell the variation principle of stationary Hamilton – Ostrogradsky is used. After standard transformations in the variation functional, taking into account the conditions of the contact shell – cross rib, we obtained two groups of equations: 1) the wave equations of smooth cylindrical shell of the elliptical cross – section; 2) the wave equations of longitudinal rib, which is located along the – axis. The initial wave equations are supplemented by the corresponding boundary and initial conditions. The numerical algorithm for solving the initial – boundary value problems is based on the integro-interpolation method of constructing difference relations along the spatial coordinate and the explicit approximation of the time coordinate. According to the initial formulation of the problem the solution is constructed in a smooth area and is pasted on the lines of discontinuity (lines of design center of gravity of the cross- ribs on the middle surface smooth shell). Results of theoretical study of the stability conditions of obtained difference equations are presented. As a numerical example, the problem of the dynamic behavior of longitudinal – cross stiffened cylindrical shell with elliptical cross section under action of a distributed pulse loading is presented. Comparison with similar results of the dynamic behavior of stiffened cylindrical shells of circular cross section is made

Big-Bang is a Boundary Condition

There is a common expectation that the big-bang singularity must be resolved in quantum gravity but it is not clear how this can be achieved. A major obstacle here is the difficulty of interpreting wave-functions in quantum gravity. The standard quantum mechanical framework requires a notion of time evolution and a proper definition of an invariant inner product having a probability interpretation, both of which are seemingly problematic in quantum gravity. We show that these two issues can actually be solved by introducing the embedding coordinates as dynamical variables \`a la Isham and Kuchar. The extended theory is identical to general relativity but has a larger group of gauge symmetries. The Wheeler-DeWitt equations describe the change of the wave-function from one arbitrary spacelike slice to another, however the constraint algebra makes this evolution purely kinematical and furthermore enforces the wave-function to be constrained in the subspace of zero-energy states. An inner product can also be introduced having all the necessary requirements. In this formalism big-bang appears as a finite field space boundary on which certain boundary conditions must be imposed for mathematical consistency. We explicitly illustrate this point both in the full theory and in the minisuperspace approximation.Comment: 16 pages, Revtex 4-

Asymptotic behaviour of cylindrical waves interacting with spinning strings

We consider a family of cylindrical spacetimes endowed with angular momentum that are solutions to the vacuum Einstein equations outside the symmetry axis. This family was recently obtained by performing a complete gauge fixing adapted to cylindrical symmetry. In the present work, we find boundary conditions that ensure that the metric arising from this gauge fixing is well defined and that the resulting reduced system has a consistent Hamiltonian dynamics. These boundary conditions must be imposed both on the symmetry axis and in the region far from the axis at spacelike infinity. Employing such conditions, we determine the asymptotic behaviour of the metric close to and far from the axis. In each of these regions, the approximate metric describes a conical geometry with a time dislocation. In particular, around the symmetry axis the effect of the singularity consists in inducing a constant deficit angle and a timelike helical structure. Based on these results and on the fact that the degrees of freedom in our family of metrics coincide with those of cylindrical vacuum gravity, we argue that the analysed set of spacetimes represent cylindrical gravitational waves surrounding a spinning cosmic string. For any of these spacetimes, a prediction of our analysis is that the wave content increases the deficit angle at spatial infinity with respect to that detected around the axis.Comment: 25 pages, accepted for publication in Classical and Quantum Gravit