Kolmogorov Petrovskii Piskunov Research Articles

Spontaneous Radiation and Quantum Dynamics of Biological Plasma

A quantum solution of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with convection and nonlinear diffusion is obtained which can provide the basis for the quantum biology and quantum microphysics equation. On this basis, quantum emission of biological systems, separate microorganisms (cells or bacteria), and dust plasma particles is investigated. The possibility arises of creating a generator of hard photons with energy higher than. Life on the Earth will probably need relic radiation, which might promote occurrence of life in the Universe.

Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method

The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Symmetry Operators of the Nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov Equation with a Quadratic Operator

A class of nonlinear symmetry operators has been constructed for the many-dimensional nonlocal Fisher- Kolmogorov-Petrovskii-Piskunov equation quadratic in independent variables and derivatives. The construction of each symmetry operator includes an interwining operator for the auxiliary linear equations and additional nonlinear algebraic conditions. Symmetry operators for the one-dimensional equation with a constant influence function have been constructed in explicit form and used to obtain a countable set of exact solutions. The symmetry properties of a differential equation (DE) are associated with transforms leaving invariant the set of solutions of the equation. We shall refer to these transforms as symmetry operators (SOs). Symmetry operators make it possible to generate new solutions of an equation by using them sequentially to act on a known solution. The solution generation procedure only illustrates the potentialities of symmetry operators, but does not exhaust them. The problem is how to find symmetry operators or other symmetry transforms in explicit form. The basic mathematical apparatus used in a comprehensive study of symmetries of differential equations is the Lie group theory of continuous transformations leaving an (ordinary or partial) differential equation invariant. The invariance group of an equation is also termed a symmetry group, or a group admitted by the equation. Seeking a symmetry group is reduced to solving a system of linear constitutive equations for an infinitesimal Lie group operator (generator). The Lie group of finite transformations constructed with a generator can be applied to any solution of the equation admitting the group, and thus generate parametric sets of new solutions to the equation from a known one. A Lie group of pointwise invariance transformations of a differential equation can be considered as a set of symmetry operators of the equation. A detailed description of the application of Lie groups to ordinary DEs can be found, for instance, in Refs. 1-3. For partial differential equations (PDEs), the application of Lie group methods leans upon a procedure of prolongation of the action of a Lie group on higher partial derivatives (1-3). The generator of a prolonged Lie group has special structure and is a base object of examination in a study of the PDE symmetry properties. For a Lie group admitted by a PDE, the generator is determined by a linear partial differential equation in the generator coefficients that specify the so-called symmetries of the equation. A set of symmetries possesses algebraic properties, which are used to examine the properties of equations and to find sets of their solutions (1-3). However, the problem of finding SOs of nonlinear equations in explicit form by direct methods not related to Lie groups is poorly studied. In general statement, this problem has no constructive solution due to significant

Pattern formation in terms of semiclassically limited distribution on lower dimensional manifolds for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation

We have investigated the pattern formation in systems described by the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation for the cases where the dimension of the pattern concentration domain is lower than that of the domain of independent variables. We have obtained a system of integro-differential equations which describe the dynamics of the concentration domain and the semiclassically limited density distribution for a pattern in the class of trajectory concentrated functions. Also, asymptotic large time solutions have been obtained that describe the semiclassically limited distribution for a quasi-steady-state pattern on the concentration manifold. The approach is illustrated by an example for which the analytical solution is in good agreement with the results of numerical calculations.

Quantum Dynamics of Biological Plasma in the External Coulomb Field

A quantum solution to the truncated Fisher–Kolmogorov–Petrovskii–Piskunov equation with Coulomb convection and linear diffusion is derived. The quantum radiation of biological systems, individual microorganisms (cells, bacteria), and dust plasma particles in the Coulomb field is studied using the foregoing solution.

Estimate of Accuracy of Solution of the Nonlocal Fisher–Kolomogorov–Petrovskii–Piskunov Equation

The discrepancy of semiclassical asymptotics for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation is investigated. It is shown that there exist values of the parameters of the system, for which the norm of the discrepancy is bounded and the accuracy of the asymptotic solution is preserved over the entire time interval, but also values of the parameters, for which the discrepancy tends to zero, and the asymptotic solution tends to the exact one.

On the Kolmogorov-Petrovskii-Piskunov equation

We prove existence and uniqueness of the solutions of Kolmogorov Petrovskii-Piskunov (KPP) equation. We study asymptotic stability and instability of the equilibrium solution u(x; t) = 0 of KPP equation with subject to the traveling wave solutions. We show that KPP equation has not got any periodic traveling wave solution. Also, we obtain some exact traveling wave solutions of KPP equation by the first integral method.

The homotopy analysis method applied to the Kolmogorov–Petrovskii–Piskunov (KPP) and fractional KPP equations

In this paper, the Homotopy analysis method (HAM) is employed to obtain the analytical/numerical solutions for linear and nonlinear Kolmogorov Petrovskii-Piskunov (KPP) and fractional KPP equations. The proposed method is a powerful and easy-to-use analytical tool for linear and nonlinear problems. This method contains the auxiliary parameter \(h\), which provides us with a simple way to adjust and control the convergence region of solution series. Some illustrative examples are presented. Moreover the use of HAM is found to be accurate, simple, convenient, flexible and computationally attractive.

Quasi-one-dimensional waves in rodent populations in heterogeneous habitats: A consequence of elevational gradients on spatio-temporal dynamics

Wave propagation can be clearly discerned in data collected on mouse populations in the Cibola National Forest (New Mexico, USA) related to seasonal changes. During an exploration of the construction of a methodology for investigations of the spread of the Hantavirus epidemic in mice we have built a system of interacting reaction diffusion equations of the Fisher–Kolmogorov–Petrovskii–Piskunov type. Although that approach has met with clear success recently in explaining Hantavirus refugia and other spatiotemporal correlations, we have discovered that certain observed features of the wave propagation observed in the data we mention are impossible to explain unless modifications are made. However, we have found that it is possible to provide a tentative explanation/description of the observations on the basis of an assumed Allee effect proposed to exist in the dynamics. Such incorporation of the Allee effect has been found useful in several of our recent investigations both of population dynamics and pattern formation and appears to be natural to the observed system. We report on our investigation of the observations with our extended theory.

Quantum dynamics of biological systems and dust plasma nanoparticles

A quantum solution of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with convection and linear diffusion is obtained which can provide the basis for the quantum biology and quantum microphysics equation. On this basis, quantum emission of biological systems, separate microorganisms (cells or bacteria), and dust plasma particles is investigated.

Approximate Solutions of Nonlinear Fractional Kolmogorov—Petrovskii—Piskunov Equations Using an Enhanced Algorithm of the Generalized Two-Dimensional Differential Transform Method

By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.

On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation

In this paper we study the existence and qualitative properties of traveling waves associated with a nonlinear flux limited partial differential equation coupled to a Fisher–Kolmogorov–Petrovskii–Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C2 classical regularity, but also the existence of discontinuous entropy traveling wave solutions.

Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

The classical group analysis approach used to study the symmetries of integro-differential equations in a semiclassical approximation is considered for a class of nearly linear integro-differential equations. In a semiclassical approximation, an original integro-differential equation leads to a finite consistent system of differential equations whose symmetries can be calculated by performing standard group analysis.The approach is illustrated by the calculation of the Lie symmetries in explicit form for a special case of the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov population equation.

How travelling waves attract the solutions of KPP-type equations

We consider in this paper a general reaction-diffusion equation of the KPP (Kolmogorov, Petrovskii, Piskunov) type, posed on an infinite cylinder. Such a model will have a family of pulsating waves of constant speed, larger than a critical speed c∗. The family of all supercritical waves attract a large class of initial data, and we try to understand how. We describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity: the solution will converge to a whole set of translates of the same wave, and we identify the convergence dynamics as that of an effective drift, around which an effective diffusion process occurs. In several nontrivial particular cases, we are able to describe the dynamics by an effective equation.

The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow

We consider in this paper the thermo-diffusive model for flame propagation, which is a reaction-diffusion equation of the KPP (Kolmogorov, Petrovskii, Piskunov) type, posed on an infinite cylinder. Such a model has a family of travelling waves of constant speed, larger than a critical speed $c_*$. The family of all supercritical waves attract a large class of initial data, and we try to understand how. We describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity: the solution will converge to a whole set of translates of the same wave, and we identify the convergence dynamics as that of an effective drift, around which an effective diffusion process occurs.

Nonlinear hydrodynamic corrections to supersonic F–KPP wave fronts

We study the hydrodynamic corrections to the dynamics and structure of an exothermic chemical wave front of Fisher–Kolmogorov–Petrovskii–Piskunov (F–KPP) type which travels in a one-dimensional gaseous medium. We show in particular that its long time dynamics, cut-off sensitivity and leading edge behavior are almost entirely controlled by the hydrodynamic front speed correction δ U h which characterizes the pushed nature of the front. Reducing the problem to an effective comoving heterogeneous F–KPP equation, we determine two analytical expressions for δ U h : an accurate one, derived from a variational method, and an approximate one, from which one can assess the δ U h sensitivity to the shear viscosity and heat conductivity of the fluid of interest.

A second-order linearized finite difference scheme for the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation

The purpose of this paper is to study the numerical simulation of the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation. After introducing a new variable, the integro-differential equation is transformed into an equivalent coupled system of first-order differential equations. A second-order accurate difference scheme is constructed for the new system of equations, which is proved to be local uncoupled by separation of variables. It is also proved that the scheme is uniquely solvable and second-order convergent in both time and space in L 2-norm. A numerical example is given to demonstrate the theoretical results.

Numerical simulation of the one-dimensional population dynamics with nonlocal competitive losses and convection

Numerical solutions of the generalized one-dimensional Fisher–Kolmogorov–Petrovskii–Piskunov equation with nonlocal competitive losses and convection are constructed. The influence function for nonlocal losses is chosen in the form of a Gaussian distribution. The effect of convection on the dynamics of the spatially inhomogeneous distribution of the population density is investigated.

Evolution of initial distributions with one and two centers in a two-dimensional model of the reaction-diffusion type with a nonlocal interaction of finite radius

Solutions of a generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation for a nonlocal interaction of finite radius have been constructed for initial conditions with one and two localization centers by using numerical methods. The dynamics depends on the choice of the equation parameters and initial conditions. The processes of formation and interaction of the rings expanding from each of the two localization centers and the formation of dissipative structures are considered.

Semiclassical approximation for the twodimensional Fisher–Kolmogorov–Petrovskii– Piskunov equation with nonlocal nonlinearity in polar coordinates

The two-dimensional Kolmogorov–Petrovskii–Piskunov–Fisher equation with nonlocal nonlinearity and axially symmetric coefficients in polar coordinates is considered. The method of separation of variables in polar coordinates and the nonlinear superposition principle proposed by the authors are used to construct the asymptotic solution of a Cauchy problem in a special class of smooth functions. The functions of this class arbitrarily depend on the angular variable and are semiclassically concentrated in the radial variable. The angular dependence of the function has been exactly taken into account in the solution. For the radial equation, the formalism of semiclassical asymptotics has been developed for the class of functions which singularly depend on an asymptotic small parameter, whose part is played by the diffusion coefficient. A dynamic system of Einstein–Ehrenfest equations (a system of equations in mean and central moments) has been derived. The evolution operator for the class of functions under consideration has been constructed in explicit form.