The Darboux-Backlund transformation for the static 2-dimensional continuum Heisenberg chain

We construct the Darboux-Backlund transformation for the sigma model describing static configurations of the 2-dimensional classical continuum Heisenberg chain. The transformation is characterized by a non-trivial normalization matrix depending on the background solution. In order to obtain the transformation we use a new, more general, spectral problem.Comment: 12 page

On analytic descriptions of two-dimensional surfaces associated with the CP^(N-1) sigma model

We study analytic descriptions of conformal immersions of the Riemann sphere S^2 into the CP^(N-1) sigma model. In particular, an explicit expression for two-dimensional (2-D) surfaces, obtained from the generalized Weierstrass formula, is given. It is also demonstrated that these surfaces coincide with the ones obtained from the Sym-Tafel formula. These two approaches correspond to parametrizations of one and the same surface in R^(N^2-1).Comment: 6 page

Finite-gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations

We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is "unpinched" to produce a new pair of branch points and therefore a solution of higher genus. We prove that every step in this process corresponds to a cabling operation on the previous curve, and we provide a labelling scheme that matches the deformation data with the knot type of the resulting filament.Comment: 33 pages, 5 figures; submitted to Journal of Nonlinear Scienc

Scalar second order evolution equations possessing an irreducible sl2_2-valued zero curvature representation

We find all scalar second order evolution equations possessing an sl$_2$-valued zero curvature representation that is not reducible to a proper subalgebra of sl$_2$. None of these zero-curvature representations admits a parameter.Comment: 10 pages, requires nath.st

Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth century geometers. Finally, we characterize the associated normal congruences

Nonlinear Dynamics of Moving Curves and Surfaces: Applications to Physical Systems

The subject of moving curves (and surfaces) in three dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a variety of physical problems in different disciplines. Making use of the underlying geometry, one can very often relate the associated evolution equations to many interesting nonlinear evolution equations, including soliton possessing nonlinear dynamical systems. Typical examples include dynamics of filament vortices in ordinary and superfluids, spin systems, phases in classical optics, various systems encountered in physics of soft matter, etc. Such interrelations between geometric evolution and physical systems have yielded considerable insight into the underlying dynamics. We present a succinct tutorial analysis of these developments in this article, and indicate further directions. We also point out how evolution equations for moving surfaces are often intimately related to soliton equations in higher dimensions.Comment: Review article, 38 pages, 7 figs. To appear in Int. Jour. of Bif. and Chao

Principles of meiotic chromosome assembly revealed in S. cerevisiae

During meiotic prophase, chromosomes organise into a series of chromatin loops emanating from a proteinaceous axis, but the mechanisms of assembly remain unclear. Here we use Saccharomyces cerevisiae to explore how this elaborate three-dimensional chromosome organisation is linked to genomic sequence. As cells enter meiosis, we observe that strong cohesin-dependent grid-like Hi-C interaction patterns emerge, reminiscent of mammalian interphase organisation, but with distinct regulation. Meiotic patterns agree with simulations of loop extrusion with growth limited by barriers, in which a heterogeneous population of expanding loops develop along the chromosome. Importantly, CTCF, the factor that imposes similar features in mammalian interphase, is absent in S. cerevisiae, suggesting alternative mechanisms of barrier formation. While grid-like interactions emerge independently of meiotic chromosome synapsis, synapsis itself generates additional compaction that matures differentially according to telomere proximity and chromosome size. Collectively, our results elucidate fundamental principles of chromosome assembly and demonstrate the essential role of cohesin within this evolutionarily conserved process

Gross-Neveu Models, Nonlinear Dirac Equations, Surfaces and Strings

Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model (GN2), and its chiral cousin, the NJL2 model, have shown that there are phases with inhomogeneous crystalline condensates. These (static) condensates can be found analytically because the relevant Hartree-Fock and gap equations can be reduced to the nonlinear Schr\"odinger equation, whose deformations are governed by the mKdV and AKNS integrable hierarchies, respectively. Recently, Thies et al have shown that time-dependent Hartree-Fock solutions describing baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation, and can be mapped directly to classical string solutions in AdS3. Here we propose a geometric perspective for this result, based on the generalized Weierstrass spinor representation for the embedding of 2d surfaces into 3d spaces, which explains why these well-known integrable systems underlie these various Gross-Neveu gap equations, and why there should be a connection to classical string theory solutions. This geometric viewpoint may be useful for higher dimensional models, where the relevant integrable hierarchies include the Davey-Stewartson and Novikov-Veselov systems.Comment: 27 pages, 1 figur

Soliton surfaces associated with CP^{N-1} sigma models

Soliton surfaces associated with CP^{N-1} sigma models are constructed using the Generalized Weierstrass and the Fokas-Gel'fand formulas for immersion of 2D surfaces in Lie algebras. The considered surfaces are defined using continuous deformations of the zero-curvature representation of the model and its associated linear spectral problem. The theoretical framework is discussed in detail and several new examples of such surfaces are presented.Comment: 14 page

Budding yeast ATM/ATR control meiotic double-strand break (DSB) levels by down-regulating Rec114, an essential component of the DSB-machinery

An essential feature of meiosis is Spo11 catalysis of programmed DNA double strand breaks (DSBs). Evidence suggests that the number of DSBs generated per meiosis is genetically determined and that this ability to maintain a pre-determined DSB level, or "DSB homeostasis", might be a property of the meiotic program. Here, we present direct evidence that Rec114, an evolutionarily conserved essential component of the meiotic DSB-machinery, interacts with DSB hotspot DNA, and that Tel1 and Mec1, the budding yeast ATM and ATR, respectively, down-regulate Rec114 upon meiotic DSB formation through phosphorylation. Mimicking constitutive phosphorylation reduces the interaction between Rec114 and DSB hotspot DNA, resulting in a reduction and/or delay in DSB formation. Conversely, a non-phosphorylatable rec114 allele confers a genome-wide increase in both DSB levels and in the interaction between Rec114 and the DSB hotspot DNA. These observations strongly suggest that Tel1 and/or Mec1 phosphorylation of Rec114 following Spo11 catalysis down-regulates DSB formation by limiting the interaction between Rec114 and DSB hotspots. We also present evidence that Ndt80, a meiosis specific transcription factor, contributes to Rec114 degradation, consistent with its requirement for complete cessation of DSB formation. Loss of Rec114 foci from chromatin is associated with homolog synapsis but independent of Ndt80 or Tel1/Mec1 phosphorylation. Taken together, we present evidence for three independent ways of regulating Rec114 activity, which likely contribute to meiotic DSBs-homeostasis in maintaining genetically determined levels of breaks