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### Open Access Journals

CPAA

We consider a quantum many-body model describing a system of
electrons interacting with themselves and hopping from one ion to another of a
one dimensional lattice. We show that the ground state energy of such system,
as a functional of the ionic configurations, has local minima in correspondence
of configurations described by smooth $\frac{\pi}{pF}$
periodic functions, if the interaction
is repulsive and large enough and pF is the Fermi momentum of the electrons.
This means physically that a $d=1$ metal develop a periodic distortion of
its reticular structure (Peierls instability). The minima are found solving the
Eulero-Lagrange equations of the energy by a contraction method.

CPAA

We prove the existence of small amplitude periodic solutions, for a
large Lebesgue measure set of frequencies, in the nonlinear beam
equation with a weak quadratic and velocity dependent nonlinearity
and with Dirichelet boundary conditions. Such nonlinear PDE can be
regarded as a simple model describing oscillations of flexible
structures like suspension bridges in presence of an uniform wind
flow. The periodic solutions are explicitly constructed by a
convergent perturbative expansion which can be considered the
analogue of the Lindstedt series expansion for the invariant tori in
classical mechanics. The periodic solutions are defined only in a
Cantor set, and resummation techniques of divergent powers series
are used in order to control the small divisors problem.

CPAA

We prove the existence of oscillatory
solutions of the nonlinear wave equation, under
irrationality conditions stronger than the usual
Diophantine one, by perturbative techniques
inspired by the Lindstedt series method originally introduced
in classical mechanics to study the existence of invariant
tori in quasi-integrable Hamiltonian systems.

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