Integrable nonlinear field equations and loop algebra structures

We apply the (direct and inverse) prolongation method to a couple of nonlinear Schr{\"o}dinger equations. These are taken as a laboratory field model for analyzing the existence of a connection between the integrability property and loop algebras. Exploiting a realization of the Kac-Moody type of the incomplete prolongation algebra associated with the system under consideration, we develop a procedure with allows us to generate a new class of integrable nonlinear field equations containing the original ones as a special case.Comment: 13 pages, latex, no figures

The Prolongation Problem for the Heavenly Equation

We provide an exact regular solution of an operator system arising as the prolongation structure associated with the heavenly equation. This solution is expressed in terms of operator Bessel coefficients.Comment: 9 pages, Proc. SIGRAV Conference (Bari 1998

"Soft" Anharmonic Vortex Glass in Ferromagnetic Superconductors

Ferromagnetic order in superconductors can induce a {\em spontaneous} vortex (SV) state. For external field ${\bf H}=0$, rotational symmetry guarantees a vanishing tilt modulus of the SV solid, leading to drastically different behavior than that of a conventional, external-field-induced vortex solid. We show that quenched disorder and anharmonic effects lead to elastic moduli that are wavevector-dependent out to arbitrarily long length scales, and non-Hookean elasticity. The latter implies that for weak external fields $H$, the magnetic induction scales {\em universally} like $B(H)\sim B(0)+ c H^{\alpha}$, with $\alpha\approx 0.72$. For weak disorder, we predict the SV solid is a topologically ordered vortex glass, in the ``columnar elastic glass'' universality class.Comment: minor corrections; version published in PR

Quaternionic eigenvalue problem

We discuss the (right) eigenvalue equation for $\mathbb{H}$, $\mathbb{C}$ and $\mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the quaternionic problem into an {\em equivalent} real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.Comment: 13 pages, AMS-Te

Quantum coherent transport in a three-arm beam splitter and a Braess paradox

The Braess paradox encountered in classical networks is a counterintuitive phenomenon when the flow in a road network can be impeded by adding a new road or, more generally, the overall net performance can degrade after addition of an extra available choice. In this work, we discuss the possibility of a similar effect in a phase-coherent quantum transport and demonstrate it by example of a simple Y-shaped metallic fork. To reveal the Braess-like partial suppression of the charge flow in such device, it is proposed to transfer two outgoing arms into a superconducting state. We show that the differential conductance-vs-voltage spectrum of the hybrid fork structure varies considerably when the extra link between the two superconducting leads is added and it can serve as an indicator of quantum correlations which manifest themselves in the quantum Braess paradox.Comment: 9 pages, 3 figures, the author version presented at the Quantum 2017 Workshop (Torino, Italy, 7-13 May 2017) and submitted to the International Journal of Quantum Information; v2: reference 9 added and the introduction extende