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Isolated and generalized isolated waves in dispersive media. (English. Russian original) Zbl 0908.76102
J. Appl. Math. Mech. 61, No. 4, 587-600 (1997); translation from Prikl. Mat. Mekh. 61, No. 4, 606-620 (1997).
The propagation of waves in dispersive medium is modelled by the system of equations \[ \begin{gathered} L(\tfrac{\partial}{\partial x}, \tfrac{\partial}{\partial t})w + F(w,\tfrac{\partial^i}{\partial x^i}w, \tfrac{\partial^i}{\partial x^i}\tfrac{\partial}{\partial t}w) = 0, \qquad i\leq r, \quad w\in \mathbb{R}^r, \\ L(\tfrac{\partial}{\partial x}, \tfrac{\partial}{\partial t})= \sum_{i=1}^rA_i \tfrac{\partial^i}{\partial x^i} + \sum_{j=1}^{r-1}B_j \tfrac{\partial^j}{\partial x^j}\tfrac{\partial}{\partial t} + C\tfrac{\partial}{\partial t}, \qquad j\leq r-1,\end{gathered} \] where \(A_i\), \(B_j\) and \(C\) are constant \(n\times n\)-matrices. The system is invariant with respect to the inversion \(t\to -t\) and \(x\to -x\). The dispersion correlations are obtained for the equation \(Lw = 0\), where \(w = c \exp\{i(kx - wt)\}\), and \(c\) is a constant vector. Additionally, the author proves the existence of plane longitudinal and generalized longitudinal waves. Some applications of the obtained results to the one-dimensional wave motion in a cold plasma are discussed.
MSC:
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35L05 Wave equation
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