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H. Mohamad and M. Oliver,
A direct construction of a slow manifold for a semilinear
wave equation of Klein-Gordon type,
J. Differential Equations 267 (2019), 1-14, doi:10.1016/j.jde.2019.01.001.
Abstract:
We prove that a semilinear wave equation, related to the semilinear
Klein--Gordon equation in the non-relativistic limit, albeit with a
nonlinearity that is Fréchet-differentiable over the complex
numbers, 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 \(H^{s-1}\times H^s\) 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.
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