Level sets as progressing waves: an example for wake-free waves in every
dimension

Wolfgang Quapp
Mathematisches Institut, University of Leipzig, PF 100920,
D-04009 Leipzig, Germany

quapp@uni-leipzig.de

and

Josep Maria Bofill
Departament de Qu'imica Org`anica, Universitat de Barcelona 
Mart'i  i Franqu`es 1, 08028 Barcelona, Spain; 
Institut de Qu'imica Te`orica i Computacional, Universitat de
Barcelona, (IQTCUB), Mart'i  i Franqu`es 1, 08028 Barcelona, Spain

jmbofill@ub.edu


Abstract:

The potential energy surface of a molecule can be decomposed into
equipotential hypersurfaces of the level sets. It is a foliation. 
The main result is that the contours are the wave fronts of a
certain hyperbolic partial differential equation, a wave equation.
It is connected with the gradient lines, 
as well as with a corresponding eikonal equation.
The energy seen as an additional coordinate plays the central role in this
treatment. 
A solution is a sharp front in the form of a delta distribution. 
We discuss Huygens' principle: 
there is no wake of the wave solution in every dimension, thus, 
the cases of odd or even dimensions are equal.


 PACS number: 00.02,    MSC numbers: 35A18, 35C07, 35L05

 Keywords:  Contours, Steepest ascent, Eikonal equation,
            Wave equation, Huygens' principle

submitted to J Math Chem ---  2013  ---