### ?

## On the general traveling wave solutions of somenonlinear diffusion equations

Journal of Physics: Conference Series. 2017. Vol. 788. P. 012033-1-012033-5.

Sinelshchikov D., Кудряшов Н. А.

We consider a family of nonlinear diffusion equations with nonlinear sources. Weassume that all nonlinearities are polynomials with respect to a dependent variable. Thetraveling wave reduction of this family of equations is an equation of the Lienard–type. Applyingrecently obtained criteria for integrability of Lienard–type equations we find some new integrablefamilies of traveling wave reductions of nonlinear diffusion equations as well as their generalanalytical solutions.

Demina M.V., Sinelshchikov D., Journal of Geometry and Physics 2021 Vol. 165 P. 104215-1-104215-12

Nonlinear oscillators described by polynomial Liénard differential equations arise in a variety of mathematical and physical applications. For a family of generalized Duffing–van der Pol oscillators we classify Darboux integrable cases and explicitly construct the corresponding generalized Darboux first integrals. We demonstrate that Darboux integrability is in strong correlation with the linearizability via the generalized ...

Added: May 27, 2021

Demina M.V., Sinelshchikov D.I., Symmetry 2019 Vol. 11 No. 11 P. 1-10

We consider a family of cubic Liénard oscillators with linear damping. Particular cases of this family of equations are abundant in various applications, including physics and biology. There are several approaches for studying integrability of the considered family of equations such as Lie point symmetries, algebraic integrability, linearizability conditions via various transformations and so on. ...

Added: November 12, 2019

Romanov A., On the Hyperbolicity Properties of Inertial Manifolds of Reaction–Diffusion Equations / Cornell University. Series math "arxiv.org". 2016. No. 1602.08953.

For 3D reaction–diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an ...

Added: June 26, 2016

Sinelshchikov D., Chaos, Solitons and Fractals 2021 Vol. 152 P. 111412-1-111412-5

We consider a family of nonlinear oscillators with quadratic damping, that generalizes the Liénard equation. We show that certain nonlocal transformations preserve autonomous invariant curves of equations from this family. Thus, nonlocal transformations can be used for extending known classification of invariant curves to the whole equivalence class of the corresponding equation, which includes non-polynomial ...

Added: September 21, 2021

Sinelshchikov D., Кудряшов Н. А., Theoretical and Mathematical Physics 2018 Vol. 196 No. 2 P. 1230-1240

We study a family of nonautonomous generalized Liénard-type equations. We consider the equivalence problem via the generalized Sundman transformations between this family of equations and type-I Painlevé–Gambier equations. As a result, we find four criteria of equivalence, which give four integrable families of Liénard-type equations. We demonstrate that these criteria can be used to construct ...

Added: February 9, 2019

Sinelshchikov D., AIMS MATHEMATICS 2021 Vol. 6 No. 11 P. 12902-12910

In this work we consider a family of cubic, with respect to the first derivative, nonlinear oscillators. We obtain the equivalence criterion for this family of equations and a non-canonical form of Ince Ⅶ equation, where as equivalence transformations we use generalized nonlocal transformations. As a result, we construct two integrable subfamilies of the considered ...

Added: September 21, 2021

Romanov A., Dynamics of Partial Differential Equations 2016 Vol. 13 No. 3 P. 263-272

For 3D reaction–diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an ...

Added: June 26, 2016

Sinelshchikov D., Physics Letters A 2020 Vol. 384 No. 26 Article 126655

In this work we consider a family of nonlinear oscillators that is cubic with respect to the first derivative. Particular members of this family of equations often appear in numerous applications. We solve the linearization problem for this family of equations, where as equivalence transformations we use generalized nonlocal transformations. We explicitly find correlations on ...

Added: June 21, 2020