Non-autonomous reductions of the KdV equation and multi-component analogs of the Painlev\'e equations P34_{34} and P3_3

We study reductions of the Korteweg--de Vries equation corresponding to stationary equations for symmetries from the noncommutative subalgebra. An equivalent system of $n$ second-order equations is obtained, which reduces to the Painlev\'e equation P$_{34}$ for $n=1$. On the singular line $t=0$, a subclass of special solutions is described by a system of $n-1$ second-order equations, equivalent to the P$_3$ equation for $n=2$. For these systems, we obtain the isomonodromic Lax pairs and B\"acklund transformations which form the group ${\mathbb Z}^n_2\times{\mathbb Z}^n$.Comment: 11 page

Planar Heterostructure Graphene -- Narrow-Gap Semiconductor -- Graphene

We investigate a planar heterostructure composed of two graphene films separated by a narrow-gap semiconductor ribbon. We show that there is no the Klein paradox when the Dirac points of the Brillouin zone of graphene are in a band gap of a narrow-gap semiconductor. There is the energy range depending on an angle of incidence, in which the above-barrier damped solution exists. Therefore, this heterostructure is a "filter" transmitting particles in a certain range of angles of incidence upon a potential barrier. We discuss the possibility of an application of this heterostructure as a "switch".Comment: 9 pages, 2 figure

Interface states in junctions of two semiconductors with intersecting dispersion curves

A novel type of shallow interface state in junctions of two semiconductors without band inversion is identified within the envelope function approximation, using the two-band model. It occurs in abrupt junctions when the interband velocity matrix elements of the two semiconductors differ and the bulk dispersion curves intersect. The in-plane dispersion of the interface state is found to be confined to a finite range of momenta centered around the point of intersection. These states turn out to exist also in graded junctions, with essentially the same properties as in the abrupt case.Comment: 1 figur

Graded associative conformal algebras of finite type

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group $\Gamma$ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group $G$ such that the identity component $G^0$ is the affine line and $G/G^0\simeq \Gamma$. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained