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Integrable Hierarchies and Information Measures
In this paper we investigate integrable models from the perspective of
information theory, exhibiting various connections. We begin by showing that
compressible hydrodynamics for a one-dimesional isentropic fluid, with an
appropriately motivated information theoretic extension, is described by a
general nonlinear Schrodinger (NLS) equation. Depending on the choice of the
enthalpy function, one obtains the cubic NLS or other modified NLS equations
that have applications in various fields. Next, by considering the integrable
hierarchy associated with the NLS model, we propose higher order information
measures which include the Fisher measure as their first member. The lowest
members of the hiearchy are shown to be included in the expansion of a
regularized Kullback-Leibler measure while, on the other hand, a suitable
combination of the NLS hierarchy leads to a Wootters type measure related to a
NLS equation with a relativistic dispersion relation. Finally, through our
approach, we are led to construct an integrable semi-relativistic NLS equation.Comment: 11 page
On the modulation instability development in optical fiber systems
Extensive numerical simulations were performed to investigate all stages of
modulation instability development from the initial pulse of pico-second
duration in photonic crystal fiber: quasi-solitons and dispersive waves
formation, their interaction stage and the further propagation. Comparison
between 4 different NLS-like systems was made: the classical NLS equation, NLS
system plus higher dispersion terms, NLS plus higher dispersion and
self-steepening and also fully generalized NLS equation with Raman scattering
taken into account. For the latter case a mechanism of energy transfer from
smaller quasi-solitons to the bigger ones is proposed to explain the dramatical
increase of rogue waves appearance frequency in comparison to the systems when
the Raman scattering is not taken into account.Comment: 9 pages, 54 figure
Model Order Reduction for Nonlinear Schr\"odinger Equation
We apply the proper orthogonal decomposition (POD) to the nonlinear
Schr\"odinger (NLS) equation to derive a reduced order model. The NLS equation
is discretized in space by finite differences and is solved in time by
structure preserving symplectic mid-point rule. A priori error estimates are
derived for the POD reduced dynamical system. Numerical results for one and two
dimensional NLS equations, coupled NLS equation with soliton solutions show
that the low-dimensional approximations obtained by POD reproduce very well the
characteristic dynamics of the system, such as preservation of energy and the
solutions
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