Dmitry I. Sinelshchikov

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.

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.

Analytical solutions for problems of bubble dynamics

Abstract Recently, an asymptotic solution of the Rayleigh equation for an empty bubble in N dimensions has been obtained. Here we give the closed-form general analytical solution of this equation. We also find the general solution of the Rayleigh equation in N dimensions for the case of a gas-filled hyperspherical bubble. In addition, we include a surface tension into consideration.

Exact solutions of equations for the Burgers hierarchy

Some classes of the rational, periodic and solitary wave solutions for the Burgers hierarchy are presented. The solutions for this hierarchy are obtained by using the generalized Cole-Hopf transformation.

On the criteria for integrability of the Liénard equation

Abstract The Lienard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for integrability of the Lienard equation using an approach based on nonlocal transformations. We also obtain some of the previously known criteria for integrability of the Lienard equation as a straightforward consequence of our approach’s application. We illustrate our results by several new examples of integrable Lienard equations.

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.

An extended equation for the description of nonlinear waves in a liquid with gas bubbles

Abstract Nonlinear waves in a liquid with gas bubbles are studied. Higher order terms with respect to the small parameter are taken into account in the derivation of the equation for nonlinear waves. A nonlinear differential equation is derived for long weakly nonlinear waves taking into consideration liquid viscosity, inter-phase heat transfer and surface tension. Additional conditions for the parameters of the equation are determined for integrability of the mathematical model. The transformation for linearization of the nonlinear equation is presented too. Some exact solutions of the nonlinear equation are found for integrable and non-integrable cases. The nonlinear waves described by the nonlinear equation are numerically investigated.

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.

Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration

Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions of nonlinear evolution equation of the fourth order are found by means of the simplest equation method. Properties of nonlinear waves in a liquid with gas bubbles are discussed.