Painlevé Test, Phase Plane Analysis and Analytical Solutions of the Chavy–Waddy–Kolokolnikov Model for the Description of Bacterial Colonies

Abstract

The Chavy–Waddy–Kolokolnikov model for the description of bacterial colonies is considered. In order to establish if the mathematical model is integrable, the Painlevé test is conducted for the nonlinear ordinary differential equation which corresponds to the fourth-order partial differential equation. The restrictions on the mathematical model parameters for ordinary differential equations to pass the Painlevé test are obtained. It is determined that the method of the inverse scattering transform does not solve the Cauchy problem for the original mathematical model, since the corresponding nonlinear ordinary differential equation passes the Painlevé test only when its solution is stationary. In the case of the stationary solution, the first integral of the equation is obtained, which makes it possible to represent the general solution in the quadrature form. The stability of the stationary points of the investigated mathematical model is carried out and their classification is proposed. Periodic and solitary stationary solutions of the Chavy–Waddy–Kolokolnikov model are constructed for various parameter values. To build analytical solutions, the method of the simplest equations is also used. The solutions, obtained in the form of a truncated expansion in powers of the logistic function, are represented as a closed formula using the formula for the Newton binomial.

Analysis of Electroencephalograms Based on the Phase Plane Method

“…The two-dimensional phase space is called the phase plane; this is a coordinate plane in which two variables (phase coordinates) are placed along the coordinate axes, which uniquely determines the state of the second-order system. The speed of movement is placed along the ordinate axis and displacement along the abscissa axis of the phase plane [8][9][10].…”

Bifurcations of Phase Portraits, Exact Solutions and Conservation Laws of the Generalized Gerdjikov–Ivanov Model

“…where C 2 is an arbitrary constant. (6) In this section, we visualize the results from the previous section by analyzing the stability of equilibrium points of the traveling-wave reduction of the explored equation and study the bifurcations of its phase portraits using the first integral (6) (see [32]). Let us write Equation ( 6) before integration in its canonical form…”

Some Latest Families of Exact Solutions to Date–Jimbo–Kashiwara–Miwa Equation and Its Stability Analysis

“…In this subsection, we demonstrate the utilization of the MK procedure. If we take an auxiliary solution for Equation (15) as follows:…”

“…Yazgan et al handled the sine-Gordon expansion method [12,13]. Ghanbari and Gomez Aguilar utilized the generalized exponential function procedure [14], Kudryashov employed the simplest equation method to the Chavy-Waddy-Kolokolnikov model [15], Sebogodi et al applied the symmetry reduction method to (2+1)-dimensional combined potential Kadomtsev-Petviashvili-B-type Kadomtsev-Petviashvili [16], Sebogadi et al used the ansatz method to obtain the traveling wave solutions of the generalized Chaffee-Infante equation in (1+3) dimensions [17], Podile et al applied the multiple exp-function technique to the e (2+1)-dimensional Hirota-Satsuma-Ito equation [18], and so on [19][20][21][22][23].…”