Orbital Magnetism in Two-dimensional Integrable Systems

We study orbital magnetism of a degenerate electron gas in a number of two-dimensional integrable systems, within linear response theory. There are three relevant energy scales: typical level spacing, the energy related to the inverse time of flight across the system, and the Fermi energy. Correspondingly, there are three distinct temperature regimes: microscopic, mesoscopic, and macroscopic. In the first two regimes there are large finite-size effects in the magnetic susceptibility, whereas in the third regime the susceptibility approaches its macroscopic value. In some cases, such as a quasi-one-dimensional strip or a harmonic confining potential, it is possible to obtain analytic expressions for the susceptibility in the entire temperature range.Comment: 28 pages, Latex, 4 Postscript figure

Effect of Multiple Scattering on the Critical Electric Field for Runaway Electrons in the Atmosphere

A simple method for taking into account the multiple Coulomb scattering in construction of a separatrix (the line separating the regions of runaway and decelerating electrons in an electric field) is described. The desired line is obtained by solving a simple transcendental equation.Comment: 3 pages, 2 figure

Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations

Analytic model for a frictional shallow-water undular bore

We use the integrable Kaup-Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by friction. This is derived in Riemann variables using a modified finite-gap integration technique for the AKNS scheme. The Whitham system is then reduced to a simple first-order differential equation which is integrated numerically to obtain an asymptotic profile of the undular bore, with the local oscillatory structure described by the periodic solution of the unperturbed Kaup-Boussinesq system. This solution of the Whitham equations is shown to be consistent with certain jump conditions following directly from conservation laws for the original system. A comparison is made with the recently studied dissipationless case for the same system, where the undular bore is unsteady.Comment: 24 page

On the temperature dependence of ballistic Coulomb drag in nanowires

We have investigated within the theory of Fermi liquid dependence of Coulomb drag current in a passive quantum wire on the applied voltage $V$ across an active wire and on the temperature $T$ for any values of $eV/k_BT$. We assume that the bottoms of the 1D minibands in both wires almost coincide with the Fermi level. We come to conclusions that 1) within a certain temperature interval the drag current can be a descending function of the temperature $T$; 2) the experimentally observed temperature dependence $T^{-0.77}$ of the drag current can be interpreted within the framework of Fermi liquid theory; 3) at relatively high applied voltages the drag current as a function of the applied voltage saturates; 4) the screening of the electron potential by metallic gate electrodes can be of importance.Comment: 7 pages, 1 figur

Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlev\'e-II equation

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order \e^2, \e\ll 1, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as $\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\epsilon^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a multiscale expansion of the solution of KdV in terms of the Hastings-McLeod solution of the Painlev\'e-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order $\epsilon^{2/3}$.Comment: 20 pages, 14 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