Pavel N. Ryabov

Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations

Abstract The application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated.

Exact solutions of the Kudryashov–Sinelshchikov equation

Abstract The Kudryashov–Sinelshchikov equation for describing the pressure waves in liquid with gas bubbles is studied. New exact solutions of this equation are found. Modification of truncated expansion method is used for obtaining exact solution of this equation.

Exact solutions of one pattern formation model

Abstract The nonlinear evolution equation for describing the pattern formation processes on the semiconductor surfaces under ion beam bombardment is studied. The Painleve analysis of equation is considered. The new elliptic solution of this equation is obtained.

A note on New kink-shaped solutions and periodic wave solutions for the (2+1)-dimensional Sine-Gordon equation

Exact solutions of the Nizhnik-Novikov-Veselov equation by Li [New kink-shaped solutions and periodic wave solutions for the (2+1)-dimensional Sine-Gordon equation, Appl. Math. Comput. 215 (2009) 3777-3781] are analyzed. We have observed that fourteen solutions by Li from 30 do not satisfy the equation. The other 16 exact solutions by Li can be found from the general solutions of the well-known solution of the equation for the Weierstrass elliptic function.

A note on the “Exp-function method for traveling waves of nonlinear evolution equations”

Abstract In this note we analyze the paper of Noor et al. (2010) [1] . Using the Exp-function method Noor et al. found the “generalized solitary and periodic solutions” of Zakharov–Kuznetsov and Zakharov–Kuznetsov-Modified-Equal-Width equations. We have checked Noor’s solutions and proved that seven from ten of them does not satisfy to equations considered. The general solution of the Zakharov–Kuznetsov equation in the traveling waves was obtained along ago. We give this solutions in this letter for the reference source.

The collective behavior of shear strain localizations in dipolar materials

Abstract We study the collective behavior of shear bands in HY-100 steel and OFHC copper taking into account the dipolar effects. Starting from mathematical model, we present new numerical methodology that allows one to simulate the processes of shear strain localization in nonpolar and dipolar materials. The verification procedure was performed to prove the efficiency and accuracy of the proposed method. Using the proposed algorithm we investigate the statistical characteristics of the shear strain localization processes in dipolar materials and compare results with nonpolar case. In particular, we obtain the statistical distributions of the width of localization zones and distance between them.

Numerical Simulation of Adiabatic Shear-Band Formation in Composites

The process of plastic flow localization in a composite material consisting of welded steel and copper plates under shear strain is considered. The mathematical model of this physical process is formulated. A new numerical algorithm based on the Courant–Isaacson–Rees scheme is proposed. The algorithm is verified using three test problems. The algorithm efficiency and performance are proved by the test simulations. The proposed algorithm is used for numerical simulation of plastic strain localization on composite materials. The influence of boundary conditions, the initial plastic strain rate, and the width of the materials forming the composite bar on the localization process is studied. It is demonstrated that at the initial stage, the shear velocity for the material layers varies. Theoretical estimates of the oscillation frequency and period are proposed; the calculations using these estimates agree completely with the numerical experiments. It is established that the deformation is localized in the copper part of the composite. One or two localization regions situated at a typical distance from the boundaries are formed, depending on the width of the steel and copper parts, as well as the initial plastic strain rate and the chosen boundary conditions. The dependence of this distance on the initial plastic strain rate is demonstrated and corresponding estimates for boundary conditions of two types are obtained. It is established that in the case of two localization regions, the temperature and deformation in one of them increase much faster than in the other, while at the initial stage these quantities are nearly equal in both regions.

Analytical and numerical solutions of the generalized dispersive Swift-Hohenberg equation

The generalization of the Swift-Hohenberg equation is studied. It is shown that the equation does not pass the Kovalevskaya test and does not possess the Painleve property. Exact solutions of the generalized Swift-Hohenberg equation which are very useful to test numerical algorithms for various boundary value problems are obtained. The numerical algorithm which is based on the Crank-Nicolson-Adams-Bashforth scheme is developed. This algorithm is tested using the exact solutions. The selforganization processes described by the generalization of the Swift-Hohenberg equation are studied.

Dissipative structures of the Kuramoto–Sivashinsky equation

The features of dissipative structure formation, which is described by the periodic boundary value problem for the Kuramoto–Sivashinsky equation, are investigated. A numerical algorithm based on the pseudospectral method is presented. The efficiency and accuracy of the proposed numerical method are proved using the exact solution of the equation under study. Using the proposed method, the process of dissipative structure formation, which is described by the Kuramoto–Sivashinsky equation, is studied. The quantitative and qualitative characteristics of this process are described. It is shown that there is a value of the control parameter for which the dissipative structure formation processes occur. Via cyclic convolution, the average value of the control parameter is found. In addition, the dependence of the amplitude of the formed structures on the value of the control parameter is analyzed.