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### Open Access Journals

DCDS

In this note, it is shown that some Hamiltonian partial
differential equations such as semi-linear Schrödinger
equations, semi-linear wave equations and semi-linear beam
equations are partially integrable, i.e., they possess integrable
invariant manifolds foliated by invariant tori which carry
periodic or quasi-periodic solutions. The linear stability of the
obtained invariant manifolds is also concluded. The proofs are
based on a special invariant property of the considered equations
and a symplectic change of variables first observed in [26].

DCDS

In this paper, we consider the analytic reducibility problem of an
analytic $d-$dimensional quasi-periodic cocycle $(\alpha,\ A)$ on
$U(n)$ where $ \alpha$ is a Diophantine vector. We prove that, if
the cocycle is conjugated to a constant cocycle $(\alpha,\ C)$ by a
measurable conjugacy $(0,\ B)$, then for almost all $C$ it is
analytically conjugated to $(\alpha,\ C)$ provided that $A$ is
sufficiently close to some constant. Moreover $B$ is actually
analytic if it is continuous.

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