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Approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation
A simple construction of inertial manifolds under time discretization
| 1. | Research Center for Applied Mathematics, Xi'an Jiaotong University, Xi'an, 710049, China |
$ T_h^0\Phi(p)=\sum_{k=1}^{\infty}R(h)^kQF(p^{-k}+\Phi(p^{-k})) $
where $p^{-k}=(S^h_\Phi)^{-k}(p)$, see [1] for detailed definition. Here we get the existence by solving the following equation about $\Phi$:
$\Phi(S_\Phi^h(p))=R(h)[\Phi(p)+hQF(p+\Phi(p))] \mbox{ for }\forall p\in PH.$
See section 1 for further explanation which describes just the invariant property of inertial manifolds. Finally we prove the $C^1$ smoothness of inertial manifolds.
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