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Bäcklund Transformations in Discrete Variational Principles for Lie-Poisson Equations

Conference paper
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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)

Abstract

We consider a dynamical system on the dual of a Lie algebra. On the dual of that algebra there is another linear Poisson structure. This system is integrable for one of the Poisson structures because it admits a suitable Lax representation. The discrete variational principle is applied to the problem given by the non-usual linear Poisson structure to obtain Lie-Poisson integrators which preserve all the Casimir functions of the system. In the 19th century Bäcklund transformations were introduced in the area of partial differential equations as transformations that map solutions to solutions. It is known that Bäcklund transformations satisfy some specific properties such as commutativity. We geometrically define Bäcklund transformations associated with the obtained Lie-Poisson integrators under some invariance assumptions. The existence of an invariant scalar product that identifies the Lie algebra and the dual of the Lie algebra allows to establish the connection with the results proved in Suris’ book on the Lie algebra. We will make clear the constructions by looking at the Toda Lattice example.

Keywords

Variational integrators Lie-Poisson equations Bäcklund transformations 

MSC 2010

37K10 65P10 70G45 70H06 

Notes

Acknowledgements

This work has been supported by a grant (010-ABEL-CM-2014A) from Iceland, Liechtenstein and Norway through the EEA Financial Mechanism, operated by Universidad Complutense de Madrid, by the ICMAT Severo Ochoa project SEV-2011-0087, MTM2013-42870-P and MTM2016-76072-P. MBL developed this research mainly at Universidad Carlos III de Madrid as Assistant Professor.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departamento de Matemática AplicadaUniversidad Politécnica de MadridMadridSpain
  2. 2.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain
  3. 3.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain

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