Coupled Schrödinger equations as a model of interchain torsional excitation transport in the DNA model

In this report we consider two weakly coupled Schrödinger equations as a model of interchain energy transport in the DNA double-helix. We employ a reduction of the Yakushevich-type model that considers the torsional dynamics of the DNA. In previous works, only small amplitude excitations and stationary dynamics were investigated, whereas we focus on the nonstationary dynamics of the double helix. In this report we consider two weakly coupled Schrödinger equations as a reduced model of interchain energy transport in the DNA double-helix torsional model. We employ a reduction of the Yakushevich-type model that considers the torsional dynamics of the DNA as effective chains of pendula. In previous works, only small amplitude excitations and stationary dynamics were investigated, whereas we focus on the nonstationary dynamics of the double helix. We consider the system to be a model of two weakly interacting DNA strands. Assuming that initially only one of the chains is excited in the form of a breather, we demonstrate the existence of an invariant that allows us to reduce the order of the problem and examine the system of the phase plane. The analysis demonstrates the utility of an analytical tool for predicting the periodic interchain excitation transitions of its localisation on one of the chains. The technique also takes into account the spreading of the excitations over time.

Piecewise linear approximation of boundary conditions and solutions in problems of 1d nonstationary dynamics of heteromodular elastic media

Рассматривается динамика одномерных деформаций разномодульной упругой среды под действием нестационарной граничной нагрузки в режиме «растяжение - сжатие». Исследуются особенности построения кусочно-линейной функции, аппроксимирующей нелинейное краевое условие. Указан критерий выбора узловых точек разбиения аппроксимирующей функции, позволяющий управлять режимами взаимодействия волновых фронтов всех типов в решении краевой задачи. Получена итерационная формула изменения скорости ударной волны в результате ее попутного столкновения с медленными фронтами предварительного растяжения, а также итерационные соотношения для построения поля перемещений на всех стадиях деформирования. The one-dimensional deformation dynamics in an elastic heteromodular medium under nonstationary boundary loading in the <tension and compression> mode is considered. The features of constructing a piecewise linear approximation of a nonlinear boundary condition are investigated. A selection criterion for the nodal partition points of the approximating function is indicated; it allows us to control the modes of collision between wave fronts of all types when the boundary value problem is solved. An iterative formula for the change in the shock wave velocity as a result of its collision with slow fronts of preliminary tension is obtained; the iterative relations for constructing the displacement field at all stages of deformation are written.

Non-stationary influence function for an unbounded anisotropic Kirchhoff-Love shell

The purpose of this article is to investigate the process of the influence of a nonstationary load on an arbitrary region of an elastic anisotropic cylindrical shell. The approach to the study of the propagation of forced transient oscillations in the shell is based on the method of the influence function, which represents normal displacements in response to the action of a single load concentrated along the coordinates. For the mathematical description of the instantaneous concentrated load, the Dirac delta functions are used. To construct the influence function, expansions in exponential Fourier series and integral Laplace and Fourier transforms are applied to the original differential equations. The original integral Laplace transform is found analytically, and for the inverse integral Fourier transform, a numerical method for integrating rapidly oscillating functions is used. The convergence of the result in the Chebyshev norm is estimated. The practical significance of the work is that the obtained results can be used by scientists or students to solve new problems of dynamics of cylindrical shells on an elastic basis under pulse loads. The found non-stationary influence function opens up possibilities for studying the stress-strain state, solving nonstationary inverse and contact problems for anisotropic shells, studying nonstationary dynamics in the case of nonzero initial conditions, and also when constructing integral equations of the boundary element method.

A Nonsmooth Extension of Samuelson’s Multiplier-Accelerator Model

Dynamical-systems approaches have historically been used in business-cycle theory to generate sustained oscillations in macroeconomic variables. We aim to contribute to this literature by extending the original Samuelson multiplier-accelerator model with a discontinuous stabilization policy in terms of government expenditure. We show that the nonsmoothness yields dynamics in terms of periodic orbits and irregular fluctuations, not found in the original Samuelson model. We also note with particular interest that our model is able to generate localized nonstationary dynamics, which is in contrast to the most standard models found in the literature.