## Solitons and shock waves in molecular chains

In this field the dynamical properties of
one dimensional discrete systems which describe molecular chains have been
studied.
In the paper [43] a new discrete version of the nonlinear
Schrödinger equation (NLS) was introduced.
This equation was reported
as "Salerno's equation" in the book
*Nonlinear Science*,
by
A.C.Scott.
From a physical point of view it represents a nonlinear
generalization of the tight-binding Schrödinger model for the dynamics of
a quasiparticle in a molecular crystal, with the nonlinear terms modeling
the interaction of the quasiparticle with the lattice.
Among the interesting properties of this equation there is the fact that it
is a q-deformation of the usual discrete, non integrable, version of the NLS,
which reduce, for a particular value of the deformation parameter to the
integrable discrete version.
In the work [49] the modulational instability
of Salerno's equation
has been studied, while in [56] it has been demonstrated
that the
presence of an electrical field in the system induces spatial oscillations of the
soliton which give rise to dynamical localization. This result suggests
an interesting analogy between the soliton dynamics and the one of an
electron
moving in a perfect crystal in presence of electric fields (Bloch
oscillations). In the paper
[56] it has been shown that
Bloch oscillations of a soliton in such system are possible also
in the presence of time dependent electric fields.
In [61]
the interaction of solitons with electric fields and
impurity centers (both of dissipative and conservative nature),
has been investigated. In particular, it was shown that
depending on the intensity of the impurity, the soliton
can be either pinned or executing oscillations around
the impurity leading to the dynamical localization phenomenon.
The existence of shock waves in discrete systems
such as chains of two-level atoms, Heisenberg ferromagnets,
nonlinear Schrödinger chains, etc., has been investigated.
In particular, in the work [64]
the existence of solitons of
bright and dark type in Salerno's equation have been considered.
The stability regions of these solutions have been studies as a function
of the parameters of the system, finding, quite surprisingly, that on
the borders of these regions anomalous dispersion is possible leading
to shock waves formation.
In the papers
[65,P12]
the dynamics of these shocks have been
characterized as
a function of the background wave-number inside the Brillouin zone.
It was shown that shock waves with rectangular wave front
followed by a train of solitons and by background radiation, can exist.
In [66] this analysis was extended
to chains of two-level atoms in
the presence of exchange and dipole-dipole interaction describing Frenkel
excitons.
In particular it was found that the exchange interaction helps the formation
of the shocks (both of bright and dark type) while the dipole-dipole
interaction works against it, making unstable the background field.
The possibility of shock waves in anisotropic Heisenberg quantum chains
was investigated in [70].
The analysis has been done
by using the coherent state representation and the stationary
phase approximation
to derive quasiclassical equation of motion. It was shown that
in this approximation values of the anisotropy parameter exist
for which anomalous dispersion and shock wave formation is possible.
From a physical point of view
these shock waves separate different magnetization regions
and therefore are relevant for the formation magnetic domains.
This analysis has been extended to the case of the pure quantum regime
in [75].

In [12] a mechanical model for a class
of "multiple"
sine-Gordon equations was introduced and the quasi-soliton
solution of the double sine-Gordon system investigated.
In particular, in the papers [19,
20]
the anharmonicity effects in the
small oscillations around the 4p-kink solutions of the double
sine-Gordon system were considered. In the papers
[4,11,
14]
it has been investigated, both from an analytical and numerical point
of view, the
effect of adiabatic perturbations on the sine-Gordon
system.

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