By Symon, K.R.; Argonne National Lab.; United States. Dept. of Energy. Office of Energy Research.; United States. Dept. of Energy. Office of Scientific and Technical Information
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Extra info for Derivation of Hamiltonians for accelerators
However careful reading shows that there too only the former statement can be justified. 1) which can be obtained from the study of the integral equation (1. 9). 1 Admissible wave speeds Our first result states that the set of speeds for which such a solution decreasing to 0 exists is either empty, or, connected and unbounded above. 1. 1) has a semi-wavefront solution with speed aD decreasing to 0, then the equation has such a solution for every wave speed a ~ aD. 18. 1) implies the existence of such a solution for another equation of the same type.
Noting that in the Burgers equation the unknown u denotes a variable which is a measure of turbulence in hydrodynamic flow, the inference of the positivity of the semi-wavefront solutions is that in it is impossible to have regions with turbulence in combination with regions where there is no dissipation of energy. 7) Modelling the motion of a foam composed of gas bubbles trapped in a liquid in one spatial dimension, the unknown in this equation denotes the liquid fraction of the foam, t time, and, x a spatial coordinate.
6(i) implies that u 2 2': 4A. 24) we may therefore suppose that A < u 2/4 without loss of generality. 29) where f31 := (u - vu 2 - 4A)/2 and f32 := (u + vu 2 - 4A)/2. Furthermore if [0,8""'() denotes its maximal interval of existence contained in [0,8), 0""'( is positive on (0,8""'(),8""'( depends continuously and monotonically on "I, and, 8""'( --7 0 as "I --7 -00.
Derivation of Hamiltonians for accelerators by Symon, K.R.; Argonne National Lab.; United States. Dept. of Energy. Office of Energy Research.; United States. Dept. of Energy. Office of Scientific and Technical Information