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61.
62.
The solitary wave and wave front are two
important behaviors of nonlinear evolution equations.
Geometrically, solitary wave and wave front are all plane curve.
In this paper, they can be represented in terms of curvature
c(s), which varies with arc length s. For solitary wave when
s → ±∞, then its curvature c(s) approaches
zero, and when s=0, the curvature c(s) reaches its maximum.
For wave front, when s → ±∞, then its
curvature c(s) approaches zero, and when s=0, the curvature
c(s) is still zero, but c'(s) ≠ 0. That is, s=0 is a
turning point. When c(s) is given, the variance at some point
(x,y) in stream line with arc length s satisfies a 2-order
linear variable-coefficient ordinary differential equation. From
this equation, it can be determined qualitatively whether the
given curvature is a solitary wave or wave front. 相似文献
63.
Two particles on the sphere leave the equator moving due south and travel at a constant and equal speed along a geodesic colliding
at the south pole. An observer who is unaware of the curvature of the space will conclude that there is an attractive force
acting between the particles. On the other hand, if particles travel at the same speed (initially parallel) along geodesics
in the hyperbolic plane, then the particle paths diverge. Imagine two particles in the hyperbolic plane that are bound together
at a constant distance with their center of mass traveling along a geodesic path at a constant velocity, then the force due
to the curvature of the space acts to break the bond and increases as a quadratic function of the velocity. We consider this
problem for the sphere and the hyperbolic plane and we give the exact formula for the apparent force between the particles.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
64.
Kayll Lake 《General Relativity and Gravitation》2004,36(5):1159-1169
The norms associated with the gradients of the two non-differential invariants of the Kerr vacuum are examined. Whereas both locally single out the horizons, their global behavior is more interesting. Both reflect the background angular momentum as the volume of space allowing a timelike gradient decreases with increasing angular momentum becoming zero in the degenerate and naked cases. These results extend directly to the Kerr-Newman geometry. 相似文献
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In this paper, we propose a nonlinear PDE model for reconstructing a regular surface from sampled data. At first, we show the existence and the uniqueness of a viscosity solution to this problem. Then we propose a numerical scheme for solving the nonlinear level set equation on unstructured triangulations adapted to the data sample. We show the consistency of this scheme. In addition, we show how to compute nodewise first and second order derivatives. Some application examples of curve or surface construction are provided to illustrate the potential and to demonstrate the accuracy of this method. 相似文献
70.