A poorly studied version of the three-component vector nonlinear Schrodinger equation, called the Kulish-Sklyanin model, is considered. Using the Dubrovin method, we obtain recurrent relations that allow us to obtain all equations from the corresponding hierarchy. The first two equations of the Kulish-Sklyanin hierarchy are found. It follows from these equations that the Kulish-Sklyanin model is associated with a rather non-trivial nonlinear interaction of the solution components. In particular, it is shown that the behavior of the third component of the solution of equations from the Kulish-Sklyanin hierarchy is completely determined by the parameters of the other two components. The equation of the spectral curve of the solution in the form of a plane wave is obtained.
In their recent works, Ablowitz and Musslimani proposed a new type of integrable nonlinear equations – nonlocal analogues of the nonlinear Schrödinger equation, the modified Korteweg-de Vries equation, and other nonlinear differential equations. In subsequent works, numerous researchers constructed the simplest soliton and rational solutions of these equations. In this paper, we construct the simplest oscillating solutions of some of the integrable nonlocal nonlinear differential equations associated to the nonlinear Schrödinger equation.
After publishing the pioneering works of Ablowitz and Musslimani, other authors also began active research on nonlocal forms of classical integrable nonlinear equations. They usually investigate particular equations, and for these equations they construct solutions that are expressed in terms of elementary functions. In present paper, we investigate one-phase elliptic solutions of all the equations from the AKNS hierarchy, including mixed ones. We also analyze the properties of spectral curves of the considered one-phase solutions in order to be able to construct multiphase algebro-geometric solutions of nonlocal forms of the AKNS hierarchy equations in the future.
The article shows the need to develop diagnostic methods for monitoring the quality of lubrication systems, which makes it possible to study the dynamic processes of contacting elements of the friction systems of instrument mechanisms, taking into account roughness parameters, the presence of local surface defects of elements and the bearing capacity of a lubricant. In the present article, a modern diagnostic model has been developed to control the quality of the processes of production and operation of friction systems of instrument assemblies. With the help of the developed model, it becomes possible to establish the relationship of diagnostic and design parameters of the mechanical system, as well as the appearance of possible local defects and lubricant state, which characterize the quality of friction systems used in many mechanical assemblies of the mechanisms of devices. The research results are shown in the form of nomograms to assess the defects of the elements of friction mechanisms of the mechanisms of the devices.
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