Communications in Mathematical Physics

, Volume 299, Issue 1, pp 129–161 | Cite as

Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

  • Jonatan Lenells
  • Gerard Misiołek
  • Feride Tiğlay
Article

Abstract

We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ ≠ 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups.

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Jonatan Lenells
    • 1
  • Gerard Misiołek
    • 2
  • Feride Tiğlay
    • 3
    • 4
  1. 1.Institut für Angewandte MathematikLeibniz Universität HannoverHannoverGermany
  2. 2.Department of MathematicsUniversity of Notre DameNotre DameUSA
  3. 3.Department of MathematicsUniversity of New OrleansNew OrleansUSA
  4. 4.Section de MathématiquesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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