592 research outputs found
Generalised Fourier Transform and Perturbations to Soliton Equations
A brief survey of the theory of soliton perturbations is presented. The focus
is on the usefulness of the so-called Generalised Fourier Transform (GFT). This
is a method that involves expansions over the complete basis of `squared
olutions` of the spectral problem, associated to the soliton equation. The
Inverse Scattering Transform for the corresponding hierarchy of soliton
equations can be viewed as a GFT where the expansions of the solutions have
generalised Fourier coefficients given by the scattering data.
The GFT provides a natural setting for the analysis of small perturbations to
an integrable equation: starting from a purely soliton solution one can
`modify` the soliton parameters such as to incorporate the changes caused by
the perturbation.
As illustrative examples the perturbed equations of the KdV hierarchy, in
particular the Ostrovsky equation, followed by the perturbation theory for the
Camassa- Holm hierarchy are presented.Comment: 20 pages, no figures, to appear in: Discrete and Continuous Dynamical
Systems
Coulomb interaction in graphene: Relaxation rates and transport
We analyze the inelastic electron-electron scattering in undoped graphene
within the Keldysh diagrammatic approach. We demonstrate that finite
temperature strongly affects the screening properties of graphene, which, in
turn, influences the inelastic scattering rates as compared to the
zero-temperature case. Focussing on the clean regime, we calculate the quantum
scattering rate which is relevant for dephasing of interference processes. We
identify an hierarchy of regimes arising due to the interplay of a plasmon
enhancement of the scattering and finite-temperature screening of the
interaction. We further address the energy relaxation and transport scattering
rates in graphene. We find a non-monotonic energy dependence of the inelastic
relaxation rates in clean graphene which is attributed to the resonant
excitation of plasmons. Finally, we discuss the temperature dependence of the
conductivity at the Dirac point in the presence of both interaction and
disorder. Our results complement the kinetic-equation and hydrodynamic
approaches for the collision-limited conductivity of clean graphene and can be
generalized to the treatment of physics of inelastic processes in strongly
non-equilibrium setups.Comment: 28 pages, 16 figure
Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schr\"odinger equation
We study various properties of solutions of an extended nonlinear
Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric
evolution problems -- including vortex filament dynamics -- and governs
propagation of short pulses in optical fibers and nonlinear metamaterials. For
the periodic initial-boundary value problem, we derive conservation laws
satisfied by local in time, weak (distributional) solutions, and
establish global existence of such weak solutions. The derivation is obtained
by a regularization scheme under a balance condition on the coefficients of the
linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS
model. Next, we investigate conditions for the existence of traveling wave
solutions, focusing on the case of bright and dark solitons. The balance
condition on the coefficients is found to be essential for the existence of
exact analytical soliton solutions; furthermore, we obtain conditions which
define parameter regimes for the existence of traveling solitons for various
linear dispersion strengths. Finally, we study the modulational instability of
plane waves of the ENLS equation, and identify important differences between
the ENLS case and the corresponding NLS counterpart. The analytical results are
corroborated by numerical simulations, which reveal notable differences between
the bright and the dark soliton propagation dynamics, and are in excellent
agreement with the analytical predictions of the modulation instability
analysis.Comment: 27 pages, 5 figures. To be published in Journal of Physics A:
Mathematical and Theoretica
Strong-field Phenomena in Periodic Systems
The advent of visible-infrared laser pulses carrying a substantial fraction
of their energy in a single field oscillation cycle has opened a new era in the
experimental investigation of ultrafast processes in semiconductors and
dielectrics (bulk as well as nanostructured), motivated by the quest for the
ultimate frontiers of electron-based signal metrology and processing. Exploring
ways to approach those frontiers requires insight into the physics underlying
the interaction of strong high-frequency (optical) fields with electrons moving
in periodic potentials. This Colloquium aims at providing this insight.
Introduction to the foundations of strong-field phenomena defines and compares
regimes of field--matter interaction in periodic systems, including (perfect)
crystals as well as optical and semiconductor superlattices, followed by a
review of recent experimental advances in the study of strong-field dynamics in
crystals and nanostructures. Avenues toward measuring and controlling
electronic processes up to petahertz frequencies are discussed
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