Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions
Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions
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DOI：
10.1007/s00220-010-1069-9
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发表时间：
2009-03
影响因子：
2.4
通讯作者：
J. Lenells;G. Misiołek;F. Tiglay
中科院分区：
文献类型：
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作者：
J. Lenells;G. Misiołek;F. Tiglay
本文研究了定义在圆上权为λ的张量密度空间上的一类方程，并引入了两个可积偏微分方程.其中一个方程与无粘Burgers方程密切相关，而另一个方程以前没有以任何形式被确定。我们提出了他们的Lax对公式，并描述了他们的双哈密顿结构。我们证明了相应的柯西问题的局部适定性，包括爆破以及整体解的存在性。此外，我们构造了“峰子”和“多峰”的解决方案，为所有λ = 0，1，和“冲击峰”的λ= 3。我们认为，有一个自然的几何框架，这些方程，包括其他著名的可积方程，这是基于V.阿诺德的方法，李群上的欧拉方程。
We study a family of equations defined on the space of tensor densities of weightλon 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 allλ≠ 0, 1, and “shock-peakons” forλ= 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.