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Journal of Mathematical Chemistry

, Volume 52, Issue 2, pp 654–664 | Cite as

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

  • Wolfgang Quapp
  • Josep Maria Bofill
Original Paper
  • 88 Downloads

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 of the wave equation can be a sharp front in the form of a delta distribution. We discuss a general Huygens’ principle: there is no wake of the wave solution in every dimension.

Keywords

Contours Steepest ascent Eikonal equation Wave equation Huygens’ principle 

Mathematics Subject Classification

35A18 35C07 35L05 

Notes

Acknowledgments

Financial support from the Spanish Ministerio de Ciencia e Innovación, DGI project CTQ2011-22505 and, in part from the Generalitat de Catalunya projects 2009SGR-1472 is fully acknowledged. It is a pleasure to thank Prof. R. Schimming for a helpful discussion to Ref. [13]. We thank M. Belger for a careful reading of the manuscript.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany
  2. 2.Departament de Química OrgànicaUniversitat de BarcelonaBarcelonaSpain
  3. 3.Institut de Química Teòrica i Computacional, Universitat de Barcelona, (IQTCUB)BarcelonaSpain

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