On a class of reductions of Manakov-Santini hierarchy connected with the interpolating system

Using Lax-Sato formulation of Manakov-Santini hierarchy, we introduce a class of reductions, such that zero order reduction of this class corresponds to dKP hierarchy, and the first order reduction gives the hierarchy associated with the interpolating system introduced by Dunajski. We present Lax-Sato form of reduced hierarchy for the interpolating system and also for the reduction of arbitrary order. Similar to dKP hierarchy, Lax-Sato equations for $L$ (Lax fuction) due to the reduction split from Lax-Sato equations for $M$ (Orlov function), and the reduced hierarchy for arbitrary order of reduction is defined by Lax-Sato equations for $L$ only. Characterization of the class of reductions in terms of the dressing data is given. We also consider a waterbag reduction of the interpolating system hierarchy, which defines (1+1)-dimensional systems of hydrodynamic type.Comment: 15 pages, revised and extended, characterization of the class of reductions in terms of the dressing data is give

Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems

Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms $\Omega_{N-1}$ of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deformations of (N-1)-dimensional linear subspaces. For the class of solutions which are Laurent polynomials in one variable these systems coincide with N-dimensional integrable systems such as Liouville equation (N=2), dispersionless Kadomtsev-Petviashvili equation (N=3), dispersionless Toda equation (N=3), Plebanski second heavenly equation (N=4) and others. Gauge invariant part of the forms $\Omega_{N-1}$ provides us with the compact form of the corresponding hierarchies. Dual quasi-linear systems associated with the projectively dual Grassmannians Gr(2,N+1) are defined via the requirement of the closedness of the dual forms $\Omega_{N-1}^{\star}$. It is shown that at N=3 the self-dual quasi-linear system, which is associated with the harmonic (closed and co-closed) form $\Omega_{2}$, coincides with the Maxwell equations for orthogonal electric and magnetic fields.Comment: 26 pages, references adde

On the state and outlooks of cooperation of the National Academy of Sciences of Ukraine with IIASA

The paper is dedicated to the analysis of the history and perspectives of one of the priorities of international cooperation of the NASU with the International Institute for Applied Systems Analysis (IIASA). A brief history of the establishment and early development of IIASA is described. Key research directions of the initial period of IIASA and input of scientists of Ukrainian institutes are mentioned. Research methodology, approaches and models developed at IIASA in cooperation with Ukrainian scientists are described. In particular, the role of Academician Glushkov and Academician Mikhalevich is well marked. Specific attention is focused on the period since 1994 leading to the independent National Member Organization of Ukraine. The role played by Academician Paton and Academician Shpak in this process is described. The existing IIASA network of research institutes and universities is also described. Important scientific and applied joint projects are listed. Existing perspectives in both scientific and policy making fields are discussed

Current sheets at three-dimensional magnetic nulls:effect of compressibility

The nature of current sheet formation in the vicinity of three-dimensional (3D) magnetic null points is investigated. The particular focus is upon the effect of the compressibility of the plasma on the qualitative and quantitative properties of the current sheet. An initially potential 3D null is subjected to shearing perturbations, as in a previous paper [Pontin et al., Phys. Plasmas, in press (2007)]. It is found that as the incompressible limit is approached, the collapse of the null point is suppressed, and an approximately planar current sheet aligned to the fan plane is present instead. This is the case regardless of whether the spine or fan of the null is sheared. Both the peak current and peak reconnection rate are reduced. The results have a bearing on previous analytical solutions for steady-state reconnection in incompressible plasmas, implying that fan current sheet solutions are dynamically accessible, while spine current sheet solutions are not.Comment: to appear in Physics of Plasmas. This version contains updated figures and references, additional discussion, and typos are fixed. This is the second in a series of papers - the first of which (by the same authors) is located at astro-ph/0701462. A version with higher quality figures can be found at http://www.maths.dundee.ac.uk/~dpontin

Magnetic structures and reorientation transitions in noncentrosymmetric uniaxial antiferromagnets

A phenomenological theory of magnetic states in noncentrosymmetric tetragonal antiferromagnets is developed, which has to include homogeneous and inhomogeneous terms (Lifshitz-invariants) derived from Dzyaloshinskii-Moriya couplings. Magnetic properties of this class of antiferromagnets with low crystal symmetry are discussed in relation to its first known members, the recently detected compounds Ba2CuGe2O7 and K2V3O8. Crystallographic symmetry and magnetic ordering in these systems allow the simultaneous occurrence of chiral inhomogeneous magnetic structures and weak ferromagnetism. New types of incommensurate magnetic structures are possible, namely, chiral helices with rotation of staggered magnetization and oscillations of the total magnetization. Field-induced reorientation transitions into modulated states have been studied and corresponding phase diagrams are constructed. Structures of magnetic defects (domain-walls and vortices) are discussed. In particular, vortices, i.e. localized non-singular line defects, are stabilized by the inhomogeneous Dzyaloshinskii-Moriya interactions in uniaxial noncentrosymmetric antiferromagnets.Comment: 18 pages RevTeX4, 13 figure

Astroglia-Derived BDNF and MSK-1 Mediate Experience- and Diet-Dependent Synaptic Plasticity

Experience- and diet-dependent regulation of synaptic plasticity can underlie beneficial effects of active lifestyle on the aging brain. Our previous results demonstrate a key role for brain-derived neurotrophic factor (BDNF) and MSK1 kinase in experience-related homeostatic synaptic scaling. Astroglia has been recently shown to release BDNF via a calcium-dependent mechanism. To elucidate a role for astroglia-derived BDNF in homeostatic synaptic plasticity in the aging brain, we explored the experience- and diet-related alterations of synaptic transmission and plasticity in transgenic mice with impairment of the BDNF/MSK1 pathway (MSK1 kinase dead knock-in mice, MSK1 KD) and impairment of glial exocytosis (dnSNARE mice). We found that prolonged tonic activation of astrocytes caused BDNF-dependent increase in the efficacy of excitatory synapses accompanied by enlargement of synaptic boutons. We also observed that exposure to environmental enrichment (EE) and caloric restriction (CR) enhanced the Ca2+ signalling in cortical astrocytes and strongly up-regulated the excitatory and down-regulated inhibitory synaptic currents in old wild-type mice, thus counterbalancing the impact of ageing on astroglial and synaptic signalling. The EE- and CR-induced up-scaling of excitatory synaptic transmission in neocortex was accompanied by the enhancement of long-term synaptic potentiation. Importantly, effects of EE and CR on synaptic transmission and plasticity was significantly reduced in the MSK1 KD and dnSNARE mice. Combined, our results suggest that astroglial release of BDNF is important for the homeostatic regulation of cortical synapses and beneficial effects of EE and CR on synaptic transmission and plasticity in aging brain