112 research outputs found
Electronic optics in graphene in the semiclassical approximation
We study above-barrier scattering of Dirac electrons by a smooth
electrostatic potential combined with a coordinate-dependent mass in graphene.
We assume that the potential and mass are sufficiently smooth, so that we can
define a small dimensionless semiclassical parameter . This electronic
optics setup naturally leads to focusing and the formation of caustics, which
are singularities in the density of trajectories. We construct a semiclassical
approximation for the wavefunction in all points, placing particular emphasis
on the region near the caustic, where the maximum of the intensity lies.
Because of the matrix character of the Dirac equation, this wavefunction
contains a nontrivial semiclassical phase, which is absent for a scalar wave
equation and which influences the focusing. We carefully discuss the three
steps in our semiclassical approach: the adiabatic reduction of the matrix
equation to an effective scalar equation, the construction of the wavefunction
using the Maslov canonical operator and the application of the uniform
approximation to the integral expression for the wavefunction in the vicinity
of a caustic. We consider several numerical examples and show that our
semiclassical results are in very good agreement with the results of
tight-binding calculations. In particular, we show that the semiclassical phase
can have a pronounced effect on the position of the focus and its intensity.Comment: 103 pages, 11 figure
Whitham systems and deformations
We consider the deformations of Whitham systems including the "dispersion
terms" and having the form of Dubrovin-Zhang deformations of Frobenius
manifolds. The procedure is connected with B.A. Dubrovin problem of
deformations of Frobenius manifolds corresponding to the Whitham systems of
integrable hierarchies. Under some non-degeneracy requirements we suggest a
general scheme of the deformation of the hyperbolic Whitham systems using the
initial non-linear system. The general form of the deformed Whitham system
coincides with the form of the "low-dispersion" asymptotic expansions used by
B.A. Dubrovin and Y. Zhang in the theory of deformations of Frobenius
manifolds.Comment: 27 pages, Late
Whitham method for Benjamin-Ono-Burgers equation and dispersive shocks in internal waves in deep fluid
The Whitham modulation equations for the parameters of a periodic solution
are derived using the generalized Lagrangian approach for the case of damped
Benjamin-Ono equation. The structure of the dispersive shock in internal wave
in deep water is considered by this method.Comment: 8 pages, 4 figure
Approximate techniques for dispersive shock waves in nonlinear media
Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained
Hard loss of stability in Painlev\'e-2 equation
A special asymptotic solution of the Painlev\'e-2 equation with small
parameter is studied. This solution has a critical point corresponding to
a bifurcation phenomenon. When the constructed solution varies slowly
and when the solution oscillates very fast. We investigate the
transitional layer in detail and obtain a smooth asymptotic solution, using a
sequence of scaling and matching procedures
Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type
We consider the special type of the field-theoretical Symplectic structures
called weakly nonlocal. The structures of this type are in particular very
common for the integrable systems like KdV or NLS. We introduce here the
special class of the weakly nonlocal Symplectic structures which we call the
weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then
the connection of such structures with the Whitham averaging method and propose
the procedure of "averaging" of the weakly nonlocal Symplectic structures. The
averaging procedure gives the weakly nonlocal Symplectic Structure of
Hydrodynamic Type for the corresponding Whitham system. The procedure gives
also the "action variables" corresponding to the wave numbers of -phase
solutions of initial system which give the additional conservation laws for the
Whitham system.Comment: 64 pages, Late
Double Beta Decay: Historical Review of 75 Years of Research
Main achievements during 75 years of research on double beta decay have been
reviewed. The existing experimental data have been presented and the
capabilities of the next-generation detectors have been demonstrated.Comment: 25 pages, typos adde
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