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 $h \ll 1$. 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

Classical and quantum dynamics of a particle in a narrow angle

We consider the 2D Schr¨odinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding classical problem is the billiard in this domain. In general, the corresponding dynamical system is not integrable. The small angle is a small parameter which allows one to make the averaging and reduce the classical dynamical system to an integrable one modulo exponential small correction. We use the quantum adiabatic approximation (operator separation of variables) to construct the asymptotic eigenfunctions (quasimodes) of the Schrodinger operator. We discuss the relation between classical averaging and constructed quasimodes. The behavior of quasimodes in the neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy functions that follows from different representations of asymptotics near the cusp

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

Asymptotic Solutions of the Wave Equation with Degenerate Velocity and with Right-Hand Side Localized in Space and Time

We study the Cauchy problem for the inhomogeneous two-dimensional wave equation with variable coefficients and zero initial data. The righthand side is assumed to be localized in space and time. The equation is considered in a domain with a boundary (shore). The velocity is assumed to vanish on the shore as a square root of the distance to the shore, that is, the wave equation has a singularity on the curve. This curve determines the boundary of the domain where the problem is studied. The main result of the paper is efficient asymptotic formulas for the solution of this problem, including the neighborhood of the shore.Вивчається задача Кошi для неоднорiдного двовимiрного хвильового рiвняння зi змiнними коефiцiєнтами та нульовими початковими даними. Вважається, що права частина локалiзована в просторi та часi. Рiвняння розглядається в областi з межею (берегом). Вважається, що швидкiсть на березi зникає як квадратний корiнь вiдстанi до берега, тобто хвильове рiвняння має задану на кривiй особливiсть. Ця крива i визначає межу областi, в якiй вивчається задача. Основний результат роботи - ефективнi асимптотичнi формули для розв’язку зазначеноЁ задачi, включаючи окiл берега

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