A concept of strong necessary conditions for extremum of functional has been applied for analysis an existence of dual equations for a system of two nonlinear Partial Differential Equations (PDE) in 1+1 dimensions. We consider two types of the dual equations: the Bäcklund transformations and the Bogomolny equations. A general form of the second order PDE with a derivative-less non-linear term has been considered. In the case of a coupled system of equations the general conditions for the existence of the Bogomolny decomposition are derived. In the case of an uncoupled system of equations the Bogomolny equations become the Bäcklund transformations. It has been found a denumerable classes of coupled systems possessing the Bogomolny relationship. Weaken the method into semi-strong necessary conditions is presented together with an application to the Lax hierarchy. The method basing on both the strong and the semi-strong necessary condition concept reduces the derivation of the dual equations to an algorithm.
Using a concept of strong necessary conditions we derive the Bogomolny decomposition for systems of two generalized elliptic and parabolic nonlinear partial differential equations (NPDE) of the second order. The generalization means that the equation coefficients depend on the field variables. According to the Cinquini-Cibrario criteria [18-20] the first type is characterized to be an elliptic, whereas the second one is a parabolic system. As a result we derive conditions for existence of the Bogomolny relationships.
Low-power hydrokinetic devices with nematic liquid crystals as the working liquid are presented. Continuous change of the effective viscosity of the nematic liquid crystals caused by application of the electric field allows for the continuous control of the coupling constant of the hydrokinetic devices.
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