The aim of this review article is to present recent advances in the theory of the dynamics of
modulated phases in the Frenkel-Kontorova model. This theory is motivated through two …

Existence of'breathers', that is, time-periodic, spatially localized solutions, is proved for a
broad range of time-reversible or Hamiltonian networks of weakly coupled oscillators. Some …

It has been proved by the principle of anti-integrable limit (renamed here anti-continuous
limit) and the implicit function theorem that, under rather weak hypotheses, there exist …

The study of one-dimensional nonrelativistic interacting Fermi systems has attracted a vast
interest over the years, among physicists and mathematicians. The mathematical interest is …

We address the issue of spatially localized periodic oscillations in coupled networks-so-
called discrete breathers-in a general context. This context is concerned with general …

We show using a series of examples that the existence of a singular limit called “anti-
integrable” can be useful for proving the existence of chaotic solutions and elucidating some …

Discrete breathers are time-periodic spatially localised motions in networks of oscillators.
Their occurrence in three principal contexts is reviewed:(i) autonomous Hamiltonian or …

Many models for structures in condensed matter can be associated with Hamiltonian
dynamical systems with discrete time. This connection is due to the fact that both models are …

We show that linear electron strings may arise in polar semiconductors. A single string
consists of M spinless fermions trapped by an extended polarization well of a cigar shape …

The spatially localized bound states of two electrons in the adiabatic two-dimensional
Holstein-Hubbard model on a square lattice are investigated both numerically and …