Alexander Makarenko (National Technical University of Ukraine, Kyiv)
Anton Popov (National Technical University of Ukraine, Kyiv)

Quasilinear hyperbolic modifications of Burgers equations as the object of symmetrical, analytical and computational investigations

Abstract:
Recent science and technologies follow to expanding the list of model equations with rich set of interesting solutions. For example in hydrodynamics with accounting memory (relaxation) effects many equations had been considered in [1]. Here as the new interesting model equation we propose quasilinear hyperbolic modification of Burgers equation (QHMB): $$ \tau\frac{\partial^2u}{\partial t^2} +\alpha\frac{\partial u}{\partial t}+\beta\varphi(u)\frac{\partial u}{\partial x}=\mu k(u)\frac{\partial^2u}{\partial x^2} +\nu\psi(u)\left(\frac{\partial u}{\partial x}\right)^2+\theta f(u), \qquad \qquad (1) $$ With different values of constants and nonlinear functions equation (1) contains many known equations: linear heat equation; Burgers equation; wave equation; Hopf equation; Klein–Gordon equation, including Liuville and sine-Gordon equations; telegraph equation; hyperbolic modification of Burgers equation; parabolic equation with growing nonlinear source etc. Equation (1) for special coefficients and nonlinear functions has a rich variety of behavior types: solitons; autowaves; blow-up solutions; chaos; periodic solutions; localized solutions; multivalued solutions; breathers etc.

In proposed talk we give review of results of some our investigation of equation (1) including nonexisting of solutions; comparisons theorems, singular perturbations. Remark that such effects are interesting for multiscale modeling. Also equation (1) is very rich object for symmetry analysis. Especial interesting is the possibilities for special solutions: compactons, pulsons, multivalued autowaves. Searching of coefficients values and nonlinear functions which follow to the given behavior of the solutions is the special king of inverse problems, which can be treated by special artificial neural networks.

[1] Danilenko V., Danevich T., Makarenko A., Skurativskyi S. Vladimirov V. Self-organization in nonlocal non-equilibrium media, S.I. Subbotin Institute of Geophysics of NAS of Ukraine, Kyiv, 2011, 333 p.