Sam Shen
Department of Mathematical Sciences
University of Alberta, Edmonton, Canada
Wu's Mass Postulate and Approximate Solutions of fKdV
Equation
Abstract
Wu's remarkable finding of upstream-advancing solitons in water flows
over a topography revived the nonlinear wave research in the 1980s. Wu
and his colleagues numerically and experimentally found that a
transicritical water flow over bump generates a train of
upstream-advancing solitons, a depression zone at the downstream of the
topography, and a wave zone further downstream. Wu attributed this
intriguing phenomenon to the solutions of several mathematical models,
including the forced Kortweg-de Vries (fKdV) equation. Wu (1987)
postulated that the excess mass of the upstream-advancing solitons
comes almost entirely from the region of surface depression (pp.81-82).
With this postulate, the depth of the downstream depression zone can be
found from the solvability condition of a boundary value problem of an
ordinary differential equation. Further, when the topography base is
relatively short compared to its height, the depression's depth can be
explicitly written as a function of the upstream flow speed and the
topography's cross-section area but not the shape. Then from Wu's
theorem of mass, momentum and energy, the approximate solutions of the
fKdV equation can be found. The secondpart of this talk is about the
epsilon-invariant theorem and infinitely many choices of epsilon values
when using fKdV as an asymptotic approximation model. It is interesting
to note that physically meaningful size of epsilon is in the range of
0.4-0.7, excluding values close to zero (Shen, 1992). Finally this talk
reports the satellite observations of the upstream-advancing solitons
in the atmosphere over Hainan Island, China (Zheng et al., 2003).
References:
1. T.Y. Wu, Generation of upstream advancing solitons by moving
disturbances. J. Fluid Mech. 184, 75-99 (1987).
2. S.S.P. Shen, Forced solitary waves and hydraulic falls in
two-layer flows. J. Fluid Mech. 234, 583-612 (1992).
3. Q. Zheng and co-authors, Evidence of upstream solitons and
downstream solitary wavetrains coexistence in the
real
atmosphere. Int. Journal of Remote Sensing 25,
4433-4440 (2004).
|