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

By Jonatan Lenells, G. Misiołek, Feride Tiglay. 2009

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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.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a4aef94b53f0bb593002da707271154ff682fd9d

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In Applied Mathematics

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  2. Ying Fu
  3. Yue Liu

In Mathematical Physics

  1. Changzheng Qu
  2. Ying Fu
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  4. Zhaoyang Yin
  5. Robin Ming Chen

In Statistical and Nonlinear Physics

  1. Changzheng Qu
  2. Ying Fu
  3. Yue Liu
  4. Zhaoyang Yin
  5. Robin Ming Chen

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