On the integrability of hyperbolic systems of riccati-type equations

We consider equations of the form U_xy = U * U_x, where U(x, y) is a function taking values in an arbitrary finite-dimensional algebra T over the field ℂ. We show that every such equation can be naturally associated with two characteristic Lie algebras, L_x and L_y. We define the notion of a ℤ-graded Lie algebraB corresponding to a given equation. We prove that for every equation under consideration, the corresponding algebraB can be taken as a direct sum of the vector spaces L_x and L_y if we define the commutators of the elements from L_x and L_y by means of the zero-curvature relations. Assuming that the algebra T has no left ideals, we classify the equations of the specified type associated with the finite-dimensional characteristic Lie algebras L_x and L_y. All of these equations are Darboux-integrable. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 261–275, November, 1997.     

Isoquinoline in the synthesis of 1-dichloromethylisoquinoline and 1-formylisoquinoline

By the reaction of isoquinoline with CCl₄ and methanol catalyzed by iron-containing compounds a synthesis was performed of 1-dichloromethylisoquinoline that was quantitatively converted into 1-formylisoquinoline by the treatment with dimethyl sulfoxide.     

Furfuryl alcohol in synthesis of levulinic acid esters and difurylmethane with Fe and Rh complexes

The possibility of synthesis of levulinic acid esters by the reaction of furfuryl alcohol with aliphatic alcohols in the system CCl₄-Fe(acac)₃ was studied.     

Synthesis of 1-benzofuran-2-carboxylates by reaction of 1-benzofuran with halomethanes and alcohols in the presence of iron compounds

Alkyl 1-benzofuran-2-carboxylates were obtained in quantitative yield by reaction of 1-benzofuran with halomethanes and alcohols in the presence of iron-containing catalysts both in the presence and in the absence of radical initiators.     

One-step synthesis of 3-dichloromethylpyridine from pyridine in the presence of iron-containing catalysts

3-Dichloromethylpyridine was synthesized by reaction of pyridine with the system MeOH-CCl₄-iron catalyst [FeBr₂, Fe₃(CO)₁₂, or iron(III) naphthenate]. Iron(II) bromide at a FeBr₂-pyridine-CCl₄-MeOH ratio of 1:100:200:200 showed the highest catalytic activity.     

Influx to a partially penetrating gallery with an impermeable barrier located symmetrically relative to the top and bottom of the stratum

The solution is presented of the problem of the flow rate of a partially penetrating gallery in a uniformly anisotropic stratum with an impermeable interstratum. The result obtained is used to calculate the limiting water-free oil flow; the effect of the barrier size is studied.The author wishes to thank I. A. Charnyi, V. A. Evdokimova, and I. N. Kochina for discussing the results of the present study.     

Symmetries of Systems of the Hyperbolic Riccati Type

Let be a Kac–Moody algebra, U(x,y) be a function defined in , and a be a constant element of . We prove that the equation U _xy = [[U,a],U _x] has two symmetry hierarchies connected by a gauge transformation. In particular, the well-known Konno equation appears in the case of the algebra . The corresponding symmetry hierarchies contain the nonlinear Schrödinger and the Heisenberg magnet equations.     

Decay of solutions of the first mixed problem for a high-order parabolic equation with minor terms

In a cylindric domain D = (0, ∞) × Ω where Ω ⊂ ℝ _n+1 is an unbounded domain, the first mixed problem for a high-order parabolic equation