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 second part 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 wave trains coexistence in the real atmosphere. Int. Journal of Remote Sensing 25, 4433-4440 (2004).