236 research outputs found
THE ASYMPTOTICS OF A SOLUTION OF THE MULTIDIMENSIONAL HEAT EQUATION WITH UNBOUNDED INITIAL DATA
For the multidimensional heat equation,Ā the long-time asymptotic approximation of the solutionĀ of the Cauchy problem is obtained in the case when the initial functionĀ grows at infinity and contains logarithmsĀ in its asymptotics.Ā In addition to natural applicationsĀ to processes of heat conduction and diffusion,Ā the investigation of the asymptotic behaviorĀ of the solution of the problem under considerationĀ is of interest for the asymptotic analysisof equations of parabolic type.Ā The auxiliary parameter methodĀ plays a decisive role in the investigation
Wave turbulence in Bose-Einstein condensates
The kinetics of nonequilibrium Bose-Einstein condensates are considered
within the framework of the Gross-Pitaevskii equation. A systematic derivation
is given for weak small-scale perturbations of a steady confined condensate
state. This approach combines a wavepacket WKB description with the weak
turbulence theory. The WKB theory derived in this paper describes the effect of
the condensate on the short-wave excitations which appears to be different from
a simple renormalization of the confining potential suggested in previous
literature.Comment: 33 pages 2 figure
EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACEāTIME
The solution of the Cauchy problemĀ for the vector Burgers equationĀ with a small parameter of dissipation in the -dimensional space-time is studied: With the help of the ColeāHopfĀ transform the exact solution and its leadingĀ asymptotic approximation, depending on six space-time scales,Ā near a singular point are found. A formula for the growth of partial derivativesĀ of the components of the vector field Ā on the time interval from the initial moment to the singular point,Ā called the formula of the gradient catastrophe, is established: The asymptotics of the solutionĀ far from the singular point,Ā involving a multistep reconstruction of the space-time scales,Ā is also obtained: u_{\nu} (\mathbf{x}, t, \varepsilon) \approx - 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \tanh \left[ \frac{x_{\nu}}{\varepsilon} \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty. $
Numerical simulation of the stress-strain state of the dental system
We present mathematical models, computational algorithms and software, which
can be used for prediction of results of prosthetic treatment. More interest
issue is biomechanics of the periodontal complex because any prosthesis is
accompanied by a risk of overloading the supporting elements. Such risk can be
avoided by the proper load distribution and prediction of stresses that occur
during the use of dentures. We developed the mathematical model of the
periodontal complex and its software implementation. This model is based on
linear elasticity theory and allows to calculate the stress and strain fields
in periodontal ligament and jawbone. The input parameters for the developed
model can be divided into two groups. The first group of parameters describes
the mechanical properties of periodontal ligament, teeth and jawbone (for
example, elasticity of periodontal ligament etc.). The second group
characterized the geometric properties of objects: the size of the teeth, their
spatial coordinates, the size of periodontal ligament etc. The mechanical
properties are the same for almost all, but the input of geometrical data is
complicated because of their individual characteristics. In this connection, we
develop algorithms and software for processing of images obtained by computed
tomography (CT) scanner and for constructing individual digital model of the
tooth-periodontal ligament-jawbone system of the patient. Integration of models
and algorithms described allows to carry out biomechanical analysis on
three-dimensional digital model and to select prosthesis design.Comment: 19 pages, 9 figure
Joint statistics of amplitudes and phases in Wave Turbulence
Random Phase Approximation (RPA) provides a very convenient tool to study the
ensembles of weakly interacting waves, commonly called Wave Turbulence. In its
traditional formulation, RPA assumes that phases of interacting waves are
random quantities but it usually ignores randomness of their amplitudes.
Recently, RPA was generalised in a way that takes into account the amplitude
randomness and it was applied to study of the higher momenta and probability
densities of wave amplitudes. However, to have a meaningful description of wave
turbulence the RPA properties assumed for the initial fields must be proven to
survive over the nonlinear evolution time, and such a proof is the main goal of
the present paper. We derive an evolution equation for the full probability
density function which contains the complete information about the joint
statistics of all wave amplitudes and phases. We show that, for any initial
statistics of the amplitudes, the phase factors remain statistically
independent uniformly distributed variables. If in addition the initial
amplitudes are also independent variables (but with arbitrary distributions)
they will remain independent when considered in small sets which are much less
than the total number of modes. However, if the size of a set is of order of
the total number of modes then the joint probability density for this set is
not factorisable into the product of one-mode probabilities. In the other
words, the modes in such a set are involved in a ``collective'' (correlated)
motion. We also study new type of correlators describing the phase statistics.Comment: 27 pages, uses feynmf packag
Dimensional Analysis and Weak Turbulence
In the study of weakly turbulent wave systems possessing incomplete
self-similarity it is possible to use dimensional arguments to derive the
scaling exponents of the Kolmogorov-Zakharov spectra, provided the order of the
resonant wave interactions responsible for nonlinear energy transfer is known.
Furthermore one can easily derive conditions for the breakdown of the weak
turbulence approximation. It is found that for incompletely self-similar
systems dominated by three wave interactions, the weak turbulence approximation
usually cannot break down at small scales. It follows that such systems cannot
exhibit small scale intermittency. For systems dominated by four wave
interactions, the incomplete self-similarity property implies that the scaling
of the interaction coefficient depends only on the physical dimension of the
system. These results are used to build a complete picture of the scaling
properties of the surface wave problem where both gravity and surface tension
play a role. We argue that, for large values of the energy flux, there should
be two weakly turbulent scaling regions matched together via a region of
strongly nonlinear turbulence.Comment: revtex4, 10 pages, 1 figur
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