Quasi invariant modified Sobolev norms for semi linear reversible PDEs - Mathematical PhysicsReport as inadecuate




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Abstract: We consider a general class of infinite dimensional reversible differentialsystems. Assuming a non resonance condition on the linear frequencies, weconstruct for such systems almost invariant pseudo norms that are closed toSobolev-like norms. This allows us to prove that if the Sobolev norm of index$s$ of the initial data $z 0$ is sufficiently small of order $\epsilon$ thenthe Sobolev norm of the solution is bounded by $2\epsilon$ during very longtime of order $\epsilon^{-r}$ with $r$ arbitrary. It turns out that thistheorem applies to a large class of reversible semi linear PDEs including thenon linear Schr\-odinger equation on the d-dimensional torus. We also apply ourmethod to a system of coupled NLS equations which is reversible but notHamiltonian.We also notice that for the same class of reversible systems we can prove aBirkhoff normal form theorem that in turn implies the same bounds on theSobolev norms. Nevertheless the technics that we use to prove the existence ofquasi invariant pseudo norms is much more simple and direct.



Author: Erwan Faou IRMAR, Benoit Grebert LMJL

Source: https://arxiv.org/







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