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

Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit

We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.Comment: 25 pages, 4 figure

A solution to the problems of cusps and rotation curves in dark matter halos in the cosmological standard model

We discuss various aspects of the inner structure formation in virialized dark matter (DM) halos that form as primordial density inhomogeneities evolve in the cosmological standard model. The main focus is on the study of central cusps/cores and of the profiles of DM halo rotation curves, problems that reveal disagreements among the theory, numerical simulations, and observations. A method that was developed by the authors to describe equilibrium DM systems is presented, which allows investigating these complex nonlinear structures analytically and relating density distribution profiles within a halo both to the parameters of the initial small-scale inhomogeneity field and to the nonlinear relaxation characteristics of gravitationally compressed matter. It is shown that cosmological random motions of matter `heat up' the DM particles in collapsing halos, suppressing cusp-like density profiles within developing halos, facilitating the formation of DM cores in galaxies, and providing an explanation for the difference between observed and simulated galactic rotation curves. The analytic conclusions obtained within this approach can be confirmed by the N-body model simulation once improved spatial resolution is achieved for central halo regions.Comment: 44 pages, 16 figures, 1 tabl

Mesoscopic fluctuations of Coulomb drag between quasi-ballistic 1D-wires

Quasiballistic 1D quantum wires are known to have a conductance of the order of 2e^2/h, with small sample-to-sample fluctuations. We present a study of the transconductance G_12 of two Coulomb-coupled quasiballistic wires, i.e., we consider the Coulomb drag geometry. We show that the fluctuations in G_12 differ dramatically from those of the diagonal conductance G_ii: the fluctuations are large, and can even exceed the mean value, thus implying a possible reversal of the induced drag current. We report extensive numerical simulations elucidating the fluctuations, both for correlated and uncorrelated disorder. We also present analytic arguments, which fully account for the trends observed numerically.Comment: 10 pages including 7 figures. Minor changes according to referee report. Accepted for PR

Current driven switching of magnetic layers

The switching of magnetic layers is studied under the action of a spin current in a ferromagnetic metal/non-magnetic metal/ferromagnetic metal spin valve. We find that the main contribution to the switching comes from the non-equilibrium exchange interaction between the ferromagnetic layers. This interaction defines the magnetic configuration of the layers with minimum energy and establishes the threshold for a critical switching current. Depending on the direction of the critical current, the interaction changes sign and a given magnetic configuration becomes unstable. To model the time dependence of the switching process, we derive a set of coupled Landau-Lifshitz equations for the ferromagnetic layers. Higher order terms in the non-equilibrium exchange coupling allow the system to evolve to its steady-state configuration.Comment: 8 pages, 2 figure. Submitted to Phys. Rev.

Singularities of bi-Hamiltonian systems

We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degeneracy of singular points of bi-Hamiltonian systems and describe their types

Identification of the bulk pairing symmetry in high-temperature superconductors: Evidence for an extended s-wave with eight line nodes

we identify the intrinsic bulk pairing symmetry for both electron and hole-doped cuprates from the existing bulk- and nearly bulk-sensitive experimental results such as magnetic penetration depth, Raman scattering, single-particle tunneling, Andreev reflection, nonlinear Meissner effect, neutron scattering, thermal conductivity, specific heat, and angle-resolved photoemission spectroscopy. These experiments consistently show that the dominant bulk pairing symmetry in hole-doped cuprates is of extended s-wave with eight line nodes, and of anisotropic s-wave in electron-doped cuprates. The proposed pairing symmetries do not contradict some surface- and phase-sensitive experiments which show a predominant d-wave pairing symmetry at the degraded surfaces. We also quantitatively explain the phase-sensitive experiments along the c-axis for both Bi_{2}Sr_{2}CaCu_{2}O_{8+y} and YBa_{2}Cu_{3}O_{7-y}.Comment: 11 pages, 9 figure

Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach

We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless equation. The sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painleve I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit.Comment: 30 page