Symmetry algebra of nonlinear integrable equations

The complete symmetry algebras of the Veselov-Novikov equation and of the modified Kadomtsev-Petviashvili equation are constructed. Some related questions are discussed.P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 34–41, April, 1993.     

Convergence to a Single Wave in the Fisher-KPP Equation

The authors study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation,with an initial condition that is a compact perturbation of a step function.A well-known result of Bramson states that,in the reference frame moving as 2t-(3/2) log t+x∞,the solution of the equation converges as t-→ +o∞ to a translate of the traveling wave corresponding to the minimal speed c* =2.The constant x∞ depends on the initial condition u(0,x).The proof is elaborate,and based on probabilistic arguments.The purpose of this paper is to provide a simple proof based on PDE arguments.     

Boundary Layers and KPP Fronts in a Cellular Flow

We study an eigenvalue problem associated with a reaction-diffusion-advection equation of the KPP type in a cellular flow. We obtain upper and lower bounds on the eigenvalues in the regime of a large flow amplitude A ≪ 1. It follows that the minimal pulsating traveling front speed c _*(A) satisfies the upper and lower bounds C ₁ A ^1/4≦ c _*(A)≦ C ₂ A ^1/4. Physically, the speed enhancement is related to the boundary layer structure of the associated eigenfunction – accordingly, we establish an “averaging along the streamlines” principle for the unique positive eigenfunction.     

Criteria for $ \sigma $ -Solvability and Meta- $ \sigma $ -Nilpotency of a Finite Group

Siberian Mathematical Journal - We study some conditions for a group to be $ \sigma $ -solvable or meta- $ \sigma $ -nilpotent.     

Exponential decay for the fragmentation or cell-division equation

We consider a classical integro-differential equation that arises in various applications as a model for cell-division or fragmentation. In biology, it describes the evolution of the density of cells that grow and divide. We prove the existence of a stable steady distribution (first positive eigenvector) under general assumptions in the variable coefficients case. We also prove the exponential convergence, for large times, of solutions toward such a steady state.     

Biomixing by Chemotaxis and Enhancement of Biological Reactions

Many phenomena in biology involve both reactions and chemotaxis. These processes can clearly influence each other, and chemotaxis can play an important role in sustaining and speeding up the reaction. However, to the best of our knowledge, the question of reaction enhancement by chemotaxis has not yet received extensive treatment either analytically or numerically. We consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction. The model is motivated, in particular, by studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We prove that in the framework of our model, chemotaxis plays a crucial role. There is a rigid limit to how much the fertilization efficiency can be enhanced if there is no chemotaxis but only advection and diffusion. On the other hand, when chemotaxis is present, the fertilization rate can be arbitrarily close to being complete provided that the chemotactic attraction is sufficiently strong. Moreover, an interesting feature of the estimates on fertilization rate and timescales in the chemotactic case is that they do not depend on the amplitude of the reaction term.     

Radiative Transport in a Periodic Structure

We derive radiative transport equations for solutions of a Schrödinger equation in a periodic structure with small random inhomogeneities. We use systematically the Wigner transform and the Bloch wave expansion. The streaming part of the radiative transport equations is determined entirely by the Bloch spectrum, and the scattering part by the random fluctuations.