Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation

We get the leading term of the Gurevich-Pitaevskii special solution to the KdV equation in the oscillation zone without using averaging methods.Comment: 13 pages, 3 figure

Autoresonance in a Dissipative System

We study the autoresonant solution of Duffing's equation in the presence of dissipation. This solution is proved to be an attracting set. We evaluate the maximal amplitude of the autoresonant solution and the time of transition from autoresonant growth of the amplitude to the mode of fast oscillations. Analytical results are illustrated by numerical simulations.Comment: 22 pages, 3 figure

Asymptotics for a special solution to the second member of the Painleve I hierarchy

We study the asymptotic behavior of a special smooth solution y(x,t) to the second member of the Painleve I hierarchy. This solution arises in random matrix theory and in the study of Hamiltonian perturbations of hyperbolic equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is known and relatively simple, but it turns out to be more subtle when x and t tend to infinity simultaneously. We distinguish a region of algebraic asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain rigorous asymptotics in both regions. We also discuss two critical transitional asymptotic regimes.Comment: 19 page

Peculiarities of the Accumulation of Mineral Elements and Nitrogen in the Assimilation Apparatus of Scots Pine

The research has been carried out in the Arkhangelsk forestry of the Arkhangelsk Region in a drained shrub-sphagnum pine forest. Census trees of medium diameter and height for the stand have been selected, from which samples of needles, both living and dying yellow ones, and shoots of different ages have been taken on model branches. Soil samples have also been taken from the 3 upper horizons. It has been established that most of all nitrogen and mineral elements accumulate in the living needles – 4.4 % on average, in the dying needles and shoots there are 37–40 % less. Among all the elements, nitrogen, potassium and calcium make up the largest proportion; while the rest of the chemical elements are several time less. In descending order of the share, they are arranged in the following sequence: magnesium, sulfur, phosphorus, manganese, silicon, iron and aluminum. The most important mineral elements are derived in significant quantities from the dying yellow needles. The content of sulfur, magnesium, phosphorus, manganese and iron decreases in them by 3.3, 2.1, 8.0, 1.5 and 1.4 times, respectively, compared to the living needles. Data on the presence of nitrogen and mineral elements in the soil horizons A0, T1 and T2 have been obtained. It has been established that some of the essential elements for the life of a tree are concentrated in the living needles in much larger quantities than in the soil: potassium, phosphorus, manganese and calcium – by 12.0, 4.0, 6.0 and 1.9, respectively. The yellow falling needles create a biological cycle of substances. The amount of nitrogen and ash constituents in these needles is proportional to their presence in the soil. In turn, the content of mineral elements in the shoots is close to their amount in the yellow falling needles, on the one hand, and on the other, depends on the age of the shoots. There is a steady trend of decreasing the content of nitrogen, potassium, phosphorus and sulfur with the age of the shoots