Kaloyan N. Vitanov

Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations

We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of Weierstrass and Jacobi. We show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions. This system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution. Any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation. The methodology is illustrated by obtaining solitary wave solutions for the generalized Korteweg-deVries equation and by obtaining solutions of the higher order Korteweg-deVries equation.

Traveling waves and statistical distributions connected to systems of interacting populations

We discuss the following two issues from the dynamics of interacting populations: *(I) density waves for the case or negligible random fluctuations of the population densities, *(II) probability distributions connected to the model equations for spatially averaged population densities for the case of significant random fluctuations of the independent quantity that can be associated with the population density. For the case of issue (I) we consider model equations containing polynomial nonlinearities. Such nonlinearities arise as a consequence of interaction among the populations (for the case of large population densities) or as a result of a Taylor series expansion (for the case of small density of interacting populations). By means of the modified method of the simplest equation we obtain exact traveling-wave solutions of the model equations and these solution. For the case of issue (II) we discuss model equations of the Fokker-Planck kind for the evolution of the statistical distributions of population densities. We derive a few stationary distributions for the population density and calculate the expected exit time associated with the extinction of the studied population.

Box model of migration channels

We discuss a mathematical model of migration channel based on the truncated Waring distribution. The truncated Waring distribution is obtained for a more general model of motion of substance through a channel containing finite number of boxes. The model is applied then for case of migrants moving through a channel consisting of finite number of countries or cities. The number of migrants in the channel strongly depends on the number of migrants that enter the channel through the country of entrance. It is shown that if the final destination country is very popular then large percentage of migrants may concentrate there.

Population dynamics in presence of state dependent fluctuations

We discuss a model of a system of interacting populations for the case when: (i) the growth rates and the coefficients of interaction among the populations depend on the populations densities; and (ii) the environment influences the growth rates and this influence can be modelled by a Gaussian white noise. The system of model equations for this case is a system of stochastic differential equations with: (i) deterministic part in the form of polynomial nonlinearities; and (ii) state-dependent stochastic part in the form of multiplicative Gaussian white noise. We discuss both the cases when the formal integration of the stochastic differential equations leads: (i) to integrals of Ito kind; or (ii) to integrals of Stratonovich kind. The systems of stochastic differential equations are reduced to the corresponding Fokker-Planck equations. For the Ito case and for the case of 1 population analytic results are obtained for the stationary p.d.f. of the population density. For the case of more than one population and for both the Ito case and Stratonovich case the detailed balance conditions are not satisfied. As a result the exact analytic solutions of the corresponding Fokker-Planck equations for the stationary p.d.f.s for the population densities are not known. We obtain approximate solutions for this case by the method of adiabatic elimination.

Integrability of Differential Equations with Fluid Mechanics Application: from Painleve Property to the Method of Simplest Equation

Abstract We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order possesses the Painleve property if the only movable singularities connected to this equation are single poles. The importance of this property can be seen from the Ablowitz-Ramani- Segur conhecture that states that a nonlinear PDE is solvable by inverse scattering transformation only if each nonlinear ODE obtained by ex- act reduction of this PDE possesses the Painleve property. The Painleve property motivated much research on obtaining exact solutions on non- linear PDEs and leaded in particular to the method of simplest equation. A version of this method called modified method of simplest equation is discussed below.