Infinite Schrödinger networks

Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, $q$-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator $\Delta$-theory and the $\Delta_q$-theory is investigated. In the $\Delta_q$-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative $q$-superharmonic functions is obtained in general case.

Experimental study and mathematical modelling of self-oscillation at the electrode-magnetic fluid interface in an electric field

The article describes a mathematical model of self-oscillation in the form of a boundary value problem for a nonlinear system of partial differential equations, with a numerical solution. The numerical results were compared to the experimental data to confirm the adequacy of the model. The model uses the classical system of differential equations of material balance, Nernst-Planck and Poisson equations without simplifications or fitting parameters. The aim of the article was to study the parameters of concentration self-oscillation in a layer of the dispersed phase particles of magnetic fluid at the interface with an electrode in an electric field. For this purpose, we developed a mathematical model, the consistency of which wasconfirmed by the corresponding physical mechanism.As a result of numerical experiments, we found the critical value of the potential jump after which self-oscillation began. We also determined the oscillation growth period and other characteristics of the process. We developed software called AutoWave01 with an intuitive user interface and advanced functionality for the study of self-oscillation in a thin layer of magnetic colloid.

A variational sheath model for gyrokinetic Vlasov-Poisson equations

We construct a stationary gyrokinetic variational model for sheaths close to the metallic wall of a magnetized plasma, following a physical extremalization principle for the natural energy. By considering a reduced set of parameters we show that our model has a unique minimal solution, and that the resulting electric potential has an infinite number of oscillations as it propagates towards the core of the plasma. We prove this result for the non linear problem and also provide a simpler analysis for a linearized problem, based on the construction of exact solutions. Some numerical illustrations show the well-posedness of the model after numerical discretization. They also exhibit the oscillating behavior.