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### Finite-dimensional dynamics and exact solutions of the Burgers-Huxley equation

Matviychuk R.

Moscow State University, Department of Physics, Department of Physical and Mathematical Methods of Control

The report presents a new approach to constructing exact solutions of the Burgers-Huxley equations.

The Burgers -- Huxley equation $u_t+uu_x=u_{xx}+f(u)$ is known in various fields of applied mathematics. For example, it describes transport processes in systems when diffusion and convection are equally important. It is usually assumed that the function $f$ is a polynomial of the second or third degree.

This allowed us to construct new exact solutions even in the cases when the equation does not have the necessary algebra of symmetries. The theory of finite-dimensional dynamics is a natural development of the theory of dynamical systems. Its methods make it possible to construct finite-dimensional submanifolds in an infinite-dimensional space among solutions of the equations. Elements of these submanifolds are numerated by the solutions of ordinary differential equations.

1. Kruglikov B.S., Lychagina O.V. Finite dimensional dynamics for Kolmogorov-Petrovsky-Piskunov equation // Lobachevskii Journal of Mathematics 2005; Vol.19. P. 13-28.

2. A.G. Kushner, V.V. Lychagin, V.N. Rubtsov. Contact geometry and nonlinear differential equations // Encyclopedia of Mathematics and Its Applications, 101. Cambridge: Cambridge University Press, xxii+496 pp., 2007.

3. Lychagin V. V., Lychagina O. V., Finite Dimensional Dynamics for Evolutionary Equations // Nonlinear Dyn. Vol. 48 (2007). P. 29-48.

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