B. Piette

Image analysis with two-dimensional continuous wavelet transform

Images may be analyzed and reconstructed with a two-dimensional (2D) continuous wavelet transform (CWT) based on the 2D Euclidean group with dilations. In this case, the wavelet transform of a 2D signal (an image) is a function of 4 parameters: two translation parameters b(x), b(y), a rotation angle theta and the usual dilation parameter a. For obvious practical reasons, two of the parameters must be fixed, either (a, theta) or (b(x), b(y)), and the WT visualized as a function of the two other ones. We discuss the general properties of the CWT and apply it, both analytically and graphically, to a number of simple geometrical objects: a line, a square, an angle, etc. For large a, the analysis detects the global shape of the objects, and smaller values of a reveal finer and finer details, in particular edges and contours. If the analyzing wavelet is oriented, like the 2D Morlet wavelet, the transform is extremely sensitive to directions: varying the angle theta uncovers the directional features of the objects, if any. The selectivity of a given wavelet is estimated from its reproducing kernel.

Static solutions of a D-dimensional modified nonlinear Schrödinger equation

We study static solutions of a D-dimensional modified nonlinear Schrodinger equation (MNLSE) which was shown to describe, in two dimensions, the self-trapped (spontaneously localized) electron states in a discrete isotropic electron–phonon lattice [1, 2]. We show that this MNLSE, unlike the conventional nonlinear Schrodinger equation, possesses static localized solutions at any dimensionality when the effective nonlinearity parameter is larger than a certain critical value which depends on the dimensionality of the system under study. We investigate various properties of the equation analytically, using scaling transformations, within the variational scheme and numerically, and show that the results of these studies agree qualitatively and quantitatively. In particular, we prove that, for various values of D, when the coupling constant is larger than a certain critical value (which depends on D), this equation has two solutions, a stable (metastable) and an unstable one. We show that the solutions can be well approximated by a Gaussian ansatz and we also show that, in two dimensions, the equation possesses solutions with a nonzero angular momentum.

Electron self-trapping in a discrete two-dimensional lattice

Abstract We study analytically and numerically the electron–phonon interaction in an isotropic two-dimensional lattice. We show that the properties of the system depend crucially on the electron–phonon coupling constant and that the system admits stationary soliton-like solutions when the coupling constant takes numerical values within some finite interval. We predict the lower critical value of the coupling constant and study some properties of the corresponding solutions. We estimate the period of oscillation of the slightly excited field configurations. We also prove that above the upper critical value of the coupling constant the regime of strong localisation (small polaron) takes place.

A modified Mottola-Wipf model with sphaleron and instanton fields

Abstract We modify the Mottola-Wipf O(3) sigma model by extending it with a Skyrme term. The resulting model supports a localised instanton solution, as well as sphaleron solution in the static limit.

Solitons in α-helical proteins

We investigate some aspects of the soliton dynamics in an alpha-helical protein macromolecule within the steric Davydov-Scott model. Our main objective is to elucidate the important role of the helical symmetry in the formation, stability, and dynamical properties of Davydovs solitons in an alpha helix. We show, analytically and numerically, that the corresponding system of nonlinear equations admits several types of stationary soliton solutions and that solitons which preserve helical symmetry are dynamically unstable: once formed, they decay rapidly when they propagate. On the other hand, the soliton which spontaneously breaks the local translational and helical symmetries possesses the lowest energy and is a robust localized entity. We also demonstrate that this soliton is the result of a hybridization of the quasiparticle states from the two lowest degenerate bands and has an inner structure which can be described as a modulated multihump amplitude distribution of excitations on individual spines. The complex and composite structure of the soliton manifests itself distinctly when the soliton is moving and some interspine oscillations take place. Such a soliton structure and the interspine oscillations have previously been observed numerically [A. C. Scott, Phys. Rev. A 26, 578 (1982)]. Here we argue that the solitons studied by Scott are hybrid solitons and that the oscillations arise due to the helical symmetry of the system and result from the motion of the soliton along the alpha helix. The frequency of the interspine oscillations is shown to be proportional to the soliton velocity.

Skyrmions and domain walls in dimensions

We study classical solutions of the vector O(3) sigma model in (2 + 1) dimensions, spontaneously broken to . The model possesses Skyrmion-type solutions as well as stable domain walls which connect different vacua. We show that different types of waves can propagate on the wall, including waves carrying a topological charge. The domain wall can also absorb Skyrmions and, under appropriate initial conditions, it is possible to emit a Skyrmion from the wall.

The Origin of Phragmoplast Asymmetry

The phragmoplast coordinates cytokinesis in plants [1]. It directs vesicles to the midzone, the site where they coalesce to form the new cell plate. Failure in phragmoplast function results in aborted or incomplete cytokinesis leading to embryo lethality, morphological defects, or multinucleate cells [2, 3]. The asymmetry of vesicular traffic is regulated by microtubules [1, 4, 5, 6], and the current model suggests that this asymmetry is established and maintained through treadmilling of parallel microtubules. However, we have analyzed the behavior of microtubules in the phragmoplast using live-cell imaging coupled with mathematical modeling and dynamic simulations and report that microtubules initiate randomly in the phragmoplast and that the majority exhibit dynamic instability with higher turnover rates nearer to the midzone. The directional transport of vesicles is possible because the majority of the microtubules polymerize toward the midzone. Here, we propose the first inclusive model where microtubule dynamics and phragmoplast asymmetry are consistent with the localization and activity of proteins known to regulate microtubule assembly and disassembly.

A Compartmental Model Analysis of Integrative and Self- Regulatory Ion Dynamics in Pollen Tube Growth

Sexual reproduction in higher plants relies upon the polarised growth of pollen tubes. The growth-site at the pollen tube tip responds to signalling processes to successfully steer the tube to an ovule. Essential features of pollen tube growth are polarisation of ion fluxes, intracellular ion gradients, and oscillating dynamics. However, little is known about how these features are generated and how they are causally related. We propose that ion dynamics in biological systems should be studied in an integrative and self-regulatory way. Here we have developed a two-compartment model by integrating major ion transporters at both the tip and shank of pollen tubes. We demonstrate that the physiological features of polarised growth in the pollen tube can be explained by the localised distribution of transporters at the tip and shank. Model analysis reveals that the tip and shank compartments integrate into a self-regulatory dynamic system, however the oscillatory dynamics at the tip do not play an important role in maintaining ion gradients. Furthermore, an electric current travelling along the pollen tube contributes to the regulation of ion dynamics. Two candidate mechanisms for growth-induced oscillations are proposed: the transition of tip membrane into shank membrane, and growth-induced changes in kinetic parameters of ion transporters. The methodology and principles developed here are applicable to the study of ion dynamics and their interactions with other functional modules in any plant cellular system.

Solitons inα-helical proteins

We investigate some aspects of the soliton dynamics in an alpha-helical protein macromolecule within the steric Davydov-Scott model. Our main objective is to elucidate the important role of the helical symmetry in the formation, stability, and dynamical properties of Davydovs solitons in an alpha helix. We show, analytically and numerically, that the corresponding system of nonlinear equations admits several types of stationary soliton solutions and that solitons which preserve helical symmetry are dynamically unstable: once formed, they decay rapidly when they propagate. On the other hand, the soliton which spontaneously breaks the local translational and helical symmetries possesses the lowest energy and is a robust localized entity. We also demonstrate that this soliton is the result of a hybridization of the quasiparticle states from the two lowest degenerate bands and has an inner structure which can be described as a modulated multihump amplitude distribution of excitations on individual spines. The complex and composite structure of the soliton manifests itself distinctly when the soliton is moving and some interspine oscillations take place. Such a soliton structure and the interspine oscillations have previously been observed numerically [A. C. Scott, Phys. Rev. A 26, 578 (1982)]. Here we argue that the solitons studied by Scott are hybrid solitons and that the oscillations arise due to the helical symmetry of the system and result from the motion of the soliton along the alpha helix. The frequency of the interspine oscillations is shown to be proportional to the soliton velocity.

Wobbles and other kink-breather solutions of the sine-Gordon model

We study various solutions of the sine-Gordon model in (1+1) dimensions. We use the Hirota method to construct some of them and then show that the wobble, discussed in detail in a recent paper by Kälberman, is one of such solutions. We concentrate our attention on a kink and its bound states with one or two breathers. We study their stability and some aspects of their scattering properties on potential wells and on fixed boundary conditions.