L. I. Mandel'shtam and the modern theory of nonlinear oscillations and waves

Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach

“…In that way, Haken [20][21][22] stressed the co-ordinate behavior of the parts of systems of different nature leading to the formation of global structures. Besides Haken's school of synergetics, the processes of selforganization are studied also the by Russian scientists (topics of nonlinear (autowave) processes) [23][24][25] as well as Brussels school of Ilya Prigorine [26,27]. Each one of the three schools arrives at the idea of self-organization from different starting points, stresses different aspects of it and, correspondingly, builds a different theoretical instrumentation.…”

“…Mandestam [23], initiated as early as in the 1930's the development of powerful apparatus for the mathematical description of nonlinear processes. Starting from the fundamental works of H. Poincare, they developed the theory of nonlinear oscillations [23][24][25] which allowed achieving the important generalization from point-like to distributed non-equilibrium media where the waves and structures, emerging in such media, are known as autowaves. The autowave (nonlinear dynamic) approach brings to the fore the nonlinear wave aspect of self-organization.…”

“…According to Brussels school [26,27], self-organization is a process of formation of dissipative structures as results of thermal nonequlibrium of the systems where it proposed a thermodynamic approach and considered open systems, exchanging energy and matter with the medium. The third, Russian school of nonlinear oscillations and waves, created by L.I Mandestam [23](initiated as early as in the 1930's) developed a powerful apparatus for the mathematical description of nonlinear processes [23][24][25]. In that way, all of these three approaches the synergetic (in narrow sense), the autowave (nonlinear dynamic), and the thermodynamic (dissipative structures) approach present peculiarities of selforganization.…”

“…In that way, it becomes an evolutionary natural science that allows talking about creation of a metalanguage in the problem of holistic understanding of various natural, technological, social, and economic notions basing on a single scientific concept. This concept gives us ability to create a new attitude to the process of integral comprehension of various sciences such as: physics, chemistry and biology, military science, economics, control theory, urban planning and population cybernetics, and cognitive sciences [23][24][25][26][27][28].…”

Quasi-Solitons in Two-Dimensional Composite Right- and Left-Handed Transmission Lines Periodically Loaded with Schottky Varactors

“…However, it is welknown that the 2D NS equation has no stable spatially localized soliton solutions because the nonlinearity surpasses the dispersion at any conditions. 5) We consider the Schottky varactors as the sources of nonlinearity. The nonlinearity by the Schottky varactors is introduced to the lines only in the restricted voltage range.…”