Group Analysis of Differential Equations and Integrable Systems − 2018

Vsevolod Vladimirov (Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków, Poland)
Sergii Skurativskyi (Subbotin Institute of Geophysics of NAS of Ukraine, Kyiv, Ukraine)

On the spectral stability of soliton-like solutions to a non-local hydrodynamic-type model

It is considered a model of nonlinear elastic substance containing cavities, microcracks or inclusions consisting of substances that differ sharply in physical properties from the base material. To describe the wave processes in such a medium, the averaged values of physical fields are used. This leads to nonlinear evolutionary PDEs, differing from the classical balance equations. Using some transformations, these equations can be presented in the Hamiltonian form: \begin{equation*}\label{baseq} \left(\begin{array}{c} w_t \\\eta_t\end{array} \right)=\partial_x \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right)\,\delta H, \end{equation*} where \[ H=\int_{-\infty}^{+\infty}{\left(\frac{1}{2}[\gamma\, w^2+\kappa\, (w_x)^2]- \int_{\eta_\infty}^{\eta}\left[p(\xi)-p(\eta_{\infty})\right]\,d\,\xi \right)\,d\,x}, \] $w=w(t,\,x),$ $\eta=\eta(t,\,x),$ $p(z)=z^{-(\nu+2)}$, $0\leq\nu$, $0<\eta_{\infty},$ $0<\gamma$, $0<\kappa$ are constants. The purpose of this report is twofold. In the first part, it is shown that the above system has invariant soliton-like solutions. The second part contains the results of an investigation of the spectral stability of the soliton-like solutions.