On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schr\"odinger equations

Using a moving space curve formalism, geometrical as well as gauge equivalence between a (2+1) dimensional spin equation (M-I equation) and the (2+1) dimensional nonlinear Schr\"odinger equation (NLSE) originally discovered by Calogero, discussed then by Zakharov and recently rederived by Strachan, have been estabilished. A compatible set of three linear equations are obtained and integrals of motion are discussed. Through stereographic projection, the M-I equation has been bilinearized and different types of solutions such as line and curved solitons, breaking solitons, induced dromions, and domain wall type solutions are presented. Breaking soliton solutions of (2+1) dimensional NLSE have also been reported. Generalizations of the above spin equation are discussed.Comment: 32 pages, no figures, accepted for publication in J. Math. Phy

Buffer Effects in Submersed Denitrifying Biofilter

The high content of nitrates in drinking water leads to serious diseases. The creation of biofiltering devices with the longest time of their operation between preventive flushes is extremely important. The purpose of this study was to investigate the features of the functioning of the developed U-shaped submersible denitrifying biofilter during its long-term operation in the piston filtration mode. The denitrification of water by using the method of displacement (piston) biofiltration in a submersible small U-shaped biofilter with immovable carriers of attached microflora in its filter load was studied. As a result, clogging of the pore space of the biofilter in the zone of excess bacterial nutrition is prevented and the vital activity of bacteria is maintained in places where there is no nutrient substrate. It has been shown that, due to adaptive mechanisms, denitrifying bacteria convert nitrate ions into gaseous nitrogen, consuming extracellular polymeric substances. The rate constants of the reaction of reduction of nitrates to molecular nitrogen in different zones of the biofilter under different filtration modes were determined. The activity of the microflora inside the biofilter quickly returns to its original level when a full-fledged external nutrition is resumed. The efficiency of nitrate to nitrogen conversion in the studied biofilter is 94.2±8.9%