Zlatinka I. Dimitrova

Modified method of simplest equation and its application to nonlinear PDEs

Abstract We search for traveling-wave solutions of the class of PDEs ∑ p = 1 N 1 A p ( Q ) ∂ p Q ∂ t p + ∑ r = 2 N 2 B r ( Q ) ∂ Q ∂ t r + ∑ s = 1 N 3 C s ( Q ) ∂ s Q ∂ x s + ∑ u = 2 N 4 D u ( Q ) ∂ Q ∂ x u + F ( Q ) = 0 where A p ( Q ) , B r ( Q ) , C s ( Q ) , D u ( Q ) and F ( Q ) are polynomials of Q. The basis of the investigation is a modification of the method of simplest equation. The equations of Bernoulli, Riccati and the extended tanh-function equation are used as simplest equations. The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto–Sivashinsky equation, reaction–diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.

Verhulst–Lotka–Volterra (VLV) model of ideological struggle

A general model for opinion formation and competition, like in ideological struggles is formulated. The underlying set is a closed one, like a country but in which the population size is variable in time. Several ideologies compete to increase their number of adepts. Such followers can be either converted from one ideology to another or become followers of an ideology though being previously ideologically-free. A reverse process is also allowed. We consider two kinds of conversion: unitary conversion, e.g. by means of mass communication tools, or binary conversion, e.g. by means of interactions between people. It is found that the steady state,when it exists, depends on the number of ideologies. Moreover when the number of ideologies increases some tension arises between them. This tension can change in the course of time. We propose to measure the ideology tensions through an appropriately defined scale index.

Adaptation and its impact on the dynamics of a system of three competing populations.

We investigate how the adaptation of the competition coefficients of the competing populations for the same limited resource influences the system dynamics in the regions of the parameter space, where chaotic motion of Shilnikov kind exists. We present results for two characteristic values of the competition coefficient adaptation factor α∗. The first value α∗=−0.05 belongs to the small interval of possible negative values of α∗. For this, α∗ a transition to chaos by period-doubling bifurcations occurs and a window of periodic motion exists between the two regions of chaotic motion. With increasing α∗, the system becomes more dissipative and the number of the windows of periodic motion increases. When α∗=1.0, a region of transient chaos is observed after the last window of periodic motion. We verify the picture of the system dynamics by power spectra, histograms and autocorrelations and calculate the Lyapunov exponents and Kaplan–Yorke dimension. Finally we discuss the eligibility of the investigated system for a topological analysis.

On nonlinear population waves

We discuss a model system of partial differential equations for description of the spatio-temporal dynamics of interacting populations. We are interested in the waves caused by migration of the populations. We assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations. For the particular case of one population and one spatial dimension the general model is reduced to analytically tractable PDE with polynomial nonlinearity up to 4th order. We investigate this particular case and obtain two kinds of solutions: (i) approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and (ii) exact solutions which describe nonlinear kink and solitary waves. In an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node. Finally we derive conditions for the asymptotic stability of the obtained solutions.

Influence of adaptation on the nonlinear dynamics of a system of competing populations

Abstract We investigate the nonlinear dynamics of a system of populations competing for the same limited resource assuming that they can adapt their growth rates and competition coefficients with respect to the number of individuals of each population. The adaptation leads to an enrichment of the nonlinear dynamics of the system which is demonstrated by a discussion of new orbits in the phase space of the system, completely dependent on the adaptation parameters, as well as by an investigation of the influence of the adaptation parameters on the dynamics of a strange attractor of the model system of ODEs.

Dynamical consequences of adaptation of the growth rates in a system of three competing populations

We investigate the nonlinear dynamics of a system of populations competing for the same limited resource for the case where each of the populations adapts its growth rate to the total number of individuals in all populations. We consider regions of parameter space where chaotic motion of the Shilnikov kind exists and present results for two characteristic values of the growth ratio adaptation factor r*: r* = -0.15 and 5. Negative r* can lead to vanishing of regions of chaotic motion and to a stabilization of a fixed point of the studied model system of differential equations. Positive r* lead to changes of the shape of the bifurcation diagrams in comparison with the bifurcation diagrams for the case without adaptation. For the case r* = 5 we observe transition to chaos by period-doubling bifurcations, windows of periodic motion between the regions of chaotic motion and a region of transient chaos after the last window of periodic motion. The Lyapunov dimension for the chaotic attractors is close to two and the Lyapunov spectrum has a structure which allows a topological analysis of the attractors of the investigated system.

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.

Application of the method of simplest equation for obtaining exact traveling-wave solutions for the extended Korteweg-de Vries equation and generalized Camassa-Holm equation

A version of the method of the simplest equation called modified method of simplest equation is applied to the extended Korteweg-de Vries equation and to generalized Camassa-Holm equation. Exact traveling wave solutions of these two nonlinear partial differential equations are obtained. The equations of Bernoulli, Riccati and the extended tanh-equation are used as simplest equations. Some of the obtained solutions correspond to surface water waves.

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.

Solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives

The method of simplest equation is applied to nonlinear PDEs.The PDEs contain monomials of odd and even order with respect to derivatives.Solitary wave solutions are studied.Numerous examples of the methodology are presented. We apply the method of simplest equation for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider first the general case of presence of monomials of the both (odd and even) grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade. The methodology is illustrated by numerous examples.