The Zakharov-Kuznetsov equation is a one-function equation of two variables x and y. For physics, x is the direction of wave propagation, and the deformation of the medium occurs along the perpendicular direction y. For example, an oscillation of the guitar string looks like the wave runs down the string, while the oscillations occur perpendicular direction relative to the run of the wave.
There are a large number of results that describe solutions of the Zakharov-Kuznetsov equations in the case when there are no constraints on y. But the question of wave propagation in the strip — when y is limited — was almost not studied until recently. And this is although such a statement of the problem has a physical meaning, and therefore potential applications.
RUDN University mathematicians dealt with the Zakharov-Kuznetsov equation in the strip. They examined three main cases — when there are no oscillations on the boundary of the strip, when there is no current on the same boundary and when the boundary conditions are periodic in structure. The latter case corresponds to the propagation of waves in a medium whose structure is periodic in x.
In all these cases mathematicians managed to prove theorems of existence and uniqueness of solutions. For systems of partial differential equations, which include the Zakharov-Kuznetsov equation, such equations are very rare.
These results are the first for solutions of the equation with initial conditions in the strip. Flat plasma flows with boundary conditions, which were considered by RUDN University scientists, can occur in physics and astrophysics.
The Zakharov-Kuznetsov equations belong to a wider category of equations known as the Korteweg-de Vries equations. In the study of this category of equations for the first time, it was possible to describe solitons — waves whose shape does not change during movement. Physicists consider solitons as a tool for modern optical data transmission systems. The study of solitons, which can arise in the Zakharov-Kuznetsov equations, is one of the options for the development of the work done by RUDN University mathematicians.
RUDN soil scientists have revealed a direct correlation between the rate of soil formation of carbon dioxide, called CO2 emissions, and the content of microbial biomass in it. It is known that CO2 emission from soil is mainly conditioned by respiration of soil microorganisms and plant roots. The more CO2 soil emits, the more microbial biomass it usually contains. It was shown that CO2 emission by chernozem of different ecosystems (or different types of land use) correlates with the content of microbial biomass, and most closely with the rate of its microbial respiration. And the soil with good microbial properties has the “best quality”, is more fertile, provides the highest yield of crops and other plant biomass.
A RUDN chemist has synthesized a catalyst for the production of gamma-valerolactone — an energy-intensive “green” biofuel. The catalyst based on zirconium dioxide and zeolite has shown high efficiency in converting the waste of wood plant materials — methyl levulinate — to gamma-valerolactone.
Biochemists from RUDN University determined which substances in peach leaves provide the antioxidant effect their extract has. They investigated the composition of the powders obtained from leaves of several varieties of peach and found that high polyphenol content correlates with antioxidant properties. The results will help start production of antioxidants from natural sources.