Publication

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

2010
Journal Articles

Abstract

We study a family of equations defined on the space of tensor densities of weight lambda 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 lambda not equal 0, 1, and "shock-peakons" for lambda = 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.

Official source

https://infoscience.epfl.ch/entities/publication/8457384b-ab8a-48e5-8ba8-56161b9f0f5e

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2010
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/8457384b-ab8a-48e5-8ba8-56161b9f0f5e

About this result

Related concepts (32)

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In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.

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

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