About twenty years ago, V. P. Maslov [1] put forward the idea that numerous quasilinear hyperbolic systems have only finite number of singular solution in general position. These solutions are shock waves, “infinitely narrow” solitons and point singularities of the type of the square root of a quadratic form. He has also stated conjecture that such solutions for shallow water equation can describe the dynamics of mesoscale vortices in the atmosphere, and trajectories of singularities correspond to trajectories of these vortices. Interesting integrability properties of such solutions were found in works [2, 3, 4, 5], where the shallow water equations on the β-plane with variable Coriolis force [6, 7] were considered :

We prove Maslov's conjecture that the structure of the type of square root of a quadratic form is the unique structure of weakly singular solutions (with a point singularity) of the shallow water equations with the properties of asymptotic self-similarity and stability. This fact plays a key role in the study of the dynamics of vortical singularities and their applications to the description of typhoon trajectories.

Experimental results illustrating a qualitative behavior of turbulent combustion of hydrogen mixtures with methane and butane in air are discussed. Turbulent burning velocities of hydrogen-air mixtures enriched with hydrocarbons are shown to vary inversely to changes in their laminar burning velocities. Correlations between the growth of laminar flamelets and turbulent burning velocities at various concentrations of heavy and light reagents are analyzed. The hypothesis assuming that the dependence of turbulent burning velocity on the molecular mixture composition is controlled by the representative chemical reaction time at highly curved portions of the name front is supported.

A comparative experimental study on the dynamics of propagating spherical laminar and turbulent flames in homogeneous gaseous mixtures is performed. It is shown that at the initial stage, when the curvature of the reaction and diffusion zone is large, the fireball growth velocity is much higher (lower) than the burning velocity of a well developed laminar flame at Lewis numbers greater (smaller) than unity. A comparison of the velocity of the initial laminar flame kernel growth as a function of the Lewis number with turbulent burning velocities measured in the same mixtures in a combustion chamber with a turbulizer suggests that mixture combustion at highly curved front areas controls the turbulent burning velocity. In compliance with the contemporary notions of the structure of well developed turbulence the observed trends are accounted for using the Ya.B. Zeldovich idea about the role of leading points in the mechanism of turbulent combustion.