Szegö三次方程

摘要：

我们考虑圆上$L^2$ Hardy空间上的以下哈密顿方程，$$i\partial_tu=Π(|u|^2u) ,$$，其中$Π$是Szegö投影。该方程可以看作是完全非色散发展方程的玩具模型。我们显示了这个方程的Lax对结构。我们证明了它在对合中存在无穷多个守恒律序列，并且它可以用一列有限维完全可积的哈密顿系统来逼近。我们建立了几个不稳定的现象，说明这个完全可积结构的退化。我们还对该系统的行波进行了分类。

英文摘要：

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i\partial_tu=Π(|u|^2u) ,$$ where $Π$ is the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating the degeneracy of this completely integrable structure. We also classify the traveling waves for this system.

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