Nikolay A. Kudryashov

Polynomials in logistic function and solitary waves of nonlinear differential equations

Properties of polynomials in logistic function are studied. It is demonstrated that these polynomials can be used for construction of exact solutions to nonlinear differential equations. Nonlinear differential equations with exact solutions in the form of polynomials in logistic function are found. It is shown there are solitary waves of nonlinear differential equations described by polynomial in logistic function with many maximum and minimum.

Analytical solutions of the Rayleigh equation for empty and gas-filled bubble

The Rayleigh equation for bubble dynamics is widely used. However, analytical solutions of this equation have not previously been obtained. Here we find closed-form general solutions of the Rayleigh equation both for an empty and gas-filled spherical bubble. We present an approach allowing us to construct exact solutions of the Rayleigh equation. We show that our solutions are useful for testing numerical algorithms.

Painlevé analysis and exact solutions of the Korteweg–de Vries equation with a source

Abstract We consider the Korteweg–de Vries equation with a source. The source depends on the solution as polynomials with constant coefficients. Using the Painleve test we show that the generalized Korteweg–de Vries equation is not integrable by the inverse scattering transform. However there are some exact solutions of the generalized Korteweg–de Vries equation for two forms of the source. We present these exact solutions.

On the connection of the quadratic Lienard equation with an equation for the elliptic functions

The quadratic Lienard equation is widely used in many applications. A connection between this equation and a linear second-order differential equation has been discussed. Here we show that the whole family of quadratic Lienard equations can be transformed into an equation for the elliptic functions. We demonstrate that this connection can be useful for finding explicit forms of general solutions of the quadratic Lienard equation. We provide several examples of application of our approach.

Extended models of non-linear waves in liquid with gas bubbles

Abstract In this work we generalize the models for non-linear waves in a gas–liquid mixture taking into account an interphase heat transfer, a surface tension and a weak liquid compressibility simultaneously at the derivation of the equations for non-linear waves. We also take into consideration high order terms with respect to the small parameter. Two new non-linear differential equations are derived for long weakly non-linear waves in a liquid with gas bubbles by the reductive perturbation method considering both high order terms with respect to the small parameter and the above-mentioned physical properties. One of these equations is the perturbation of the Burgers equation and corresponds to main influence of dissipation on non-linear waves propagation. The other equation is the perturbation of the Burgers–Korteweg–de Vries equation and corresponds to main influence of dispersion on non-linear waves propagation.

Quasi-exact solutions of the dissipative Kuramoto-Sivashinsky equation

The dissipative Kuramoto-Sivashinsky equation is studied. It is shown that this equation does not pass the Painleve test and as consequence this equation is not integrable. Quasi-exact solution of the dissipative Kuramoto-Sivashinsky equation is given.

Equation for the three-dimensional nonlinear waves in liquid with gas bubbles

Nonlinear waves in a liquid containing gas bubbles are considered in the three-dimensional (3D) case. The nonlinear evolution equation is given for a description of long nonlinear pressure waves. It is shown that in the general case the equation is not integrable. Some exact solutions for the nonlinear evolution equation are presented. The application of the Hirota method is illustrated for finding multi-soliton solutions for the nonintegrable evolution equation in the 3D case. The stability of the 1D solitary waves is investigated. It is shown that the 1D solitary waves are stable to transverse perturbations.

Quasi-exact solutions of nonlinear differential equations

Abstract The concept of quasi-exact solutions of nonlinear differential equations is introduced. Quasi-exact solution expands the idea of exact solution for additional values of parameters of differential equation. These solutions are approximate ones of nonlinear differential equations but they are close to exact solutions. Quasi-exact solutions of the the Kuramoto–Sivashinsky, the Korteweg-de Vries–Burgers and the Kawahara equations are founded.

Analytical solutions of the Lorenz system

The Lorenz system is considered. The Painlevé test for the third-order equation corresponding to the Lorenz model at σ ≠ 0 is presented. The integrable cases of the Lorenz system and the first integrals for the Lorenz system are discussed. The main result of the work is the classification of the elliptic solutions expressed via the Weierstrass function. It is shown that most of the elliptic solutions are degenerated and expressed via the trigonometric functions. However, two solutions of the Lorenz system can be expressed via the elliptic functions.

Self-organization of adiabatic shear bands in OFHC copper and HY-100 steel

Abstract We study the self-organization process of adiabatic shear bands in OFHC copper and HY-100 steel taking into account strain hardening factor. Starting from mathematical model we present a new numerical approach, which is based on Courant–Isaacson–Rees scheme, that allows one to simulate fully localized plastic flow. To prove the accuracy and efficiency of the following method we give solutions of two benchmark problems. Next we apply the proposed method to investigate such quantitative characteristics of self-organization process of ASB as average stress, temperature, localization time and distance between ASB. Then we compare the obtained results with theoretical predictions by other authors.