A direct construction of a slow manifold for a semilinear wave equation of Klein–Gordon type
A direct construction of a slow manifold for a semilinear wave equation of Klein–Gordon type
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DOI：
10.1016/j.jde.2019.01.001
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发表时间：
2019-06
影响因子：
2.4
通讯作者：
Haidar Mohamad;M. Oliver
中科院分区：
文献类型：
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作者：
Haidar Mohamad;M. Oliver
我们研究了一个半线性波动方程，其线性部分对应于非相对论极限下的线性 Klein-Gordon 方程，并增强了复数上 Fréchet 可微的非线性。我们证明该方程在相空间中具有几乎不变的流形，它概括了系统的有限维伽辽金截断中已知存在的慢流形。该流形对于任何代数阶几乎都是不变的，并且可以在方程的 H s− 1× H s 相空间中以近似阶一致地构造。特别是，我们证明了这个“慢流形”的动力学在有限的时间间隔内影响了整个系统的轨道。
We study a semilinear wave equation whose linear part corresponds to the linear Klein–Gordon equation in the non-relativistic limit, augmented with a nonlinearity that is Fréchet-differentiable over the complex numbers. We show that this equation possesses an almost invariant manifold in phase space that generalizes the slow manifold which is known to exist for finite-dimensional Galerkin truncations of the system. This manifold is shown to be almost invariant to any algebraic order and can be constructed in the H s− 1× H s phase space of the equation uniformly in the order of the approximation. In particular, we prove that the dynamics on this “slow manifold” shadows orbits of the full system over a finite interval of time.