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The work deals with a generalised Timoshenko type system whose coefficients are supposed to be smooth functions depending on the spatial variable only.The aim of this study focuses on establishing some sufficient condition for well-posedness of the problem and proving periodicity behavior of the existing solution when the system takes into account the influence of an elastic medium. The technique, we develop in order to derive the aforementioned results, consists of introducing some Sobolev spaces and studying the properties of the system operator, when it is embedded in them. Moreover, there is a natural comparison between the local density of the energy conservation law of the physical clamped Timoshenko beam system and the correspondent inner product of the basic pivot space, where we are interested in studying the main properties of the abstract system operator. In this way, it is proven that the system operator is maximal monotone onto the basic pivot space. Moreover, it turns out to be an anti-self-adjoint one. As a consequence, it is proven that the whole spectrum of the regarded operator consists only of completely imaginary isolated points with a unique accumulation point. [ABSTRACT FROM AUTHOR]

The thermodynamic Casimir force in a fluid system depends on the conditions the bounding surfaces exert on either the confined fluid, or binary liquid mixture. This influence is described by imposing proper boundary conditions on the order parameter describing the fluid. We present a brief review and some new results on the force dependence on the example of the Ginzburg-Landau Φ4 model via exact results and series expansions of the corresponding order-parameter profiles. We consider the (+, +), (+, −), Dirichlet-Dirichlet and Neumann-Neumann boundary conditions and a combination of them. We focus on a system with a film geometry close to its bulk critical point in which the order parameter satisfies one of the above mentioned boundary conditions. Solving the corresponding boundary-value problem of one nonlinear differential equation in terms of Weierstrass and Jacobi elliptic functions we report analytic representation of the Casimir force. We study the behavior of the force depending on both the temperature and an external ordering field acting on the system. Our results are in a full agreement with the general arguments of the finite-size scaling theory. We confirm the expectation that the force is attractive if the boundary conditions are the same on each boundary and repulsive otherwise. [ABSTRACT FROM AUTHOR]

The cell membrane is a dynamic and complex barrier which separates the living cell from its environment. It consists of a high variety of lipids and proteins and is continuously reorganized by the cell, making its in vivo study difficult. Thus one has to use membrane model systems with precisely controlled composition to investigate fundamental interactions of membrane components under well-defined conditions. Giant unilamellar vesicles (GUVs) are well-known model systems, especially because they are easily observable using optical microscopy. In the present article we describe in details the experimental setup, applied techniques and some preliminary results in the process of electroformation of giant unilamellar phospholipid vesicles (liposomes). As a base of our studies we used the protocol reported by Pereno et al [1]. [ABSTRACT FROM AUTHOR]

The Boussinesq equation with linear restoring force models the transverse or longitudinal deflection of an elastic rod subject to a constant tangential follower force and laying on a uniform elastic foundation ofWinkler type. In this paper we compute solitary waves to this equation applying the method proposed by Petviashvili. We test the accuracy of the computations on nested grids. We perform several simulations to analyze the impact of the parameters of the problem on the shape and the support of the solitary waves. [ABSTRACT FROM AUTHOR]

In this work, we study a critical thermodynamic system, say, a simple fluid or a binary liquid mixture, of plane film geometry whose stable states, at given temperature and external ordering field, are determined by the minimizers of the one-dimensional counterpart of the standard ∅4 Ginzburg-Landau Hamiltonian in terms of the order parameter. We focus on the case of Dirichlet-Dirichlet boundary conditions on the confining the fluid walls. Assuming that the boundaries of the system are positioned at a finite distance from one another, we solve the corresponding boundary-value problem of one nonlinear differential equation in terms of Weierstrass and Jacobi elliptic functions and give analytic representation of the order parameter profiles and of the local and total susceptibilities depending on the temperature and ordering field. [ABSTRACT FROM AUTHOR]

We consider a two-parameter family of nonlinear ordinary differential equations describing the behavior of a critical thermodynamic system, e.g., a binary liquid mixture, of film geometry within the framework of the Ginzburg-Landau theory by means of the order-parameter. We focus on the case in which the confining surfaces are strongly adsorbing but prefer different components of the mixture, i.e., the order-parameter tends to infinity at one of the boundaries and to minus infinity at the other one. We assume that the boundaries of the system are positioned at a finite distance from each other and give analytic solutions to the corresponding boundary-value problems in terms of Weierstrass and Jacobi elliptic functions. [ABSTRACT FROM AUTHOR]

In the current article we study the behavior of the van der Waals force between a planar substrate and an axisymmetric bilayer lipid membrane undergoing pearling instability, caused by uniform hydrostatic pressure di_erence. To do so, the recently suggested "surface integration approach" is used, which can be considered a generalization of the well known and widely used Derjaguin approximation. The static equilibrium shape after the occurrence of the instability is described in the framework of Helfrich's spontaneous curvature model. Some specific classes of exact analytical solutions to the corresponding shape equation are considered, and the components of the respective position vectors given in terms of elliptic integrals and Jacobi elliptic functions. The mutual orientation between the interacting objects is chosen such that the axis of revolution of the distorted cylinder be parallel to the plane bounding the substrate. Based on the discussed models and approaches we made some estimations for the studied force in real experimentally realizable systems, thus showing the possibility of pearling as an useful technique for reduction of the adhesion in variety of industrial processes using lipid membranes as carriers. [ABSTRACT FROM AUTHOR]

The dependence of the critical Casimir force on two parameters (temperature and external ordering field) within the mean-field Ginzburg-Landau Ising type model is studied. Here, the case of a film geometry where one boundary of the film exhibits strong adsorption to one of the phases (components) of the system, whereas the first derivative of the order-parameter profile vanishes on the other one is studied. The results obtained herein are valid for both, simple fluids and binary liquid mixtures. In the present contribution, the order parameter profile and the Casimir force are presented in exact analytic form, expressed through the Weierstrass function. Using these results, the dependence of the critical Casimir force on the temperature and external ordering field are illustrated. [ABSTRACT FROM AUTHOR]

The Willmore surfaces provide local extrema to the so-called Willmore functional, which assigns to each surface its total squared mean curvature. There is a special class of axially symmetric Willmore surfaces whose profile curves satisfy a relatively simpler system of ordinary differential equations. The aim of this work is to give explicit parametrizations of the Willmore surfaces belonging to the foregoing family in terms of the Jacobian elliptic functions and elliptic integrals. [ABSTRACT FROM AUTHOR]

The one-dimensional quasi-stationary flow of inviscid liquid in a shallow layer with free surface is described by the so-called Boussinesq Paradigm Equation (BPE). Slightly generalized this equation appears also in the theory of longitudinal vibrations of rods and in the continuum limit for lattices. It is well known that the one-dimensional (1-D) BPE admits a one-parameter family of traveling wave solutions expressed in an analytic form through the "sech" function. In the present contribution, new analytic solutions to the 1-D BPE representing traveling waves are obtained. These solutions are expressed through Weierstrass and Jacobi elliptic functions, which in some cases reduce to elementary functions. [ABSTRACT FROM AUTHOR]

The Gardner equation is well-known in the mathematical literature since the late sixties of 20th century. Initially, it appeared in the context of the construction of local conservation laws admitted by the KdV equation. Later on, the Gardner equation was generalized and found to be applicable in various branches of physics (solid-state and plasma physics, fluid dynamics and quantum field theory). In this paper, we examine the travelling wave solutions of the Gardner equation and derive the full set of solutions to the corresponding reduced equation in terms of Weierstrass and Jacobi elliptic functions. Then, we use the travelling wave solutions of the focusing mKdV equation and obtain in explicit analytic form exact solutions of a special type of plane curve flow, known as the mKdV flow. [ABSTRACT FROM AUTHOR]

The consideration of the variational problem for minimization of the bending energy of membrane surfaces under some natural geometric constraints leads to the so called general shape equation. The available analytical solutions of this equation including a new one are reviewed in some detail. [ABSTRACT FROM AUTHOR]

Starting from first principles, the curvature of the plane elastic line is expressed in terms of the slope of its tangent. This is used further for finding its intrinsic equation and for deriving a new explicit parametrization of the Euler elastica. [ABSTRACT FROM AUTHOR]