The reduction procedure for the general coupled nonlinear Schrödinger (GCNLS) equations with four-wave mixing terms is proposed. It is shown that the GCNLS system is equivalent to the well known integrable families of the Manakov and Makhankov U(n,m)-vector models. This equivalence allows us to construct bright-bright and dark-dark solitons and a quasibreather-dark solution with unconventional dynamics: the density of the first component oscillates in space and time, whereas the density of the second component does not. The collision properties of solitons are also studied.
Nanotechnology clusters found in Russia are studied. The overview of innovation regional programs is presented. It is shown that institutionally Russian regions are not ready for modern technologies implementation. The main ideas of regional experts on perspectives and tendencies in nanotechnology are reviewed. The mathematical model that describes the dynamics of innovative clusters is presented. Its application in some cases in regard to nanotechnology is shown. The issue of deliberate changes in society development (that is equivalent to changes of model parameters) by appropriate measures is discussed.
The paper proposes advancement of a stratum demographic model characterized by the presence of two strata — the Center and the Periphery, which are different in respects of their way of life, development level and social values. The processes of convergence and depopulation in the heterogeneous system of the Center-Periphery type were studied. Various variants of “catch-up development” of the Periphery are considered within the framework of numerical simulation. Modes of economic growth that contribute to the convergence of income levels in the countries of the Center and the Periphery are indicated, depending on the value of the characteristic time of convergence. There are identified the regimes of a stable and supposedly irreversible depopulation of the Periphery. It is shown that economic participation of the Center in changing the situation in the Periphery countries might be necessary to achieve economic convergence for maintaining the stability of the entire system. As an explanatory principle of the observed phenomena, the institutional trap concept is proposed. A number of socio-economic interpretations of the dynamics of the Center-Periphery system as well as the possibility of the System behavior control by appropriate management decisions are discussed on the results of computer modeling. Further development of the proposed model may include studying other scenarios of economic interaction and taking into account additional demographic and migration parameters in the equations of economic growth.
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