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On an application of the Maslov operator method to a diffraction problem. (English. Russian original) Zbl 0788.35024
Sov. Math., Dokl. 43, No. 2, 547-549 (1991); translation from Dokl. Akad. Nauk SSSR 317, No. 4, 832-834 (1991).
An application to microwave diffraction problems of the noncommutative operator calculus introduced by V. P. Maslov is studied. As is shown, this method enables uniform asymptotics to be obtained (in particular, on the boundary of the light-shadow region), and also provides a rigorous foundation for the semiheuristic method associated with the use of “diffraction rays” introduced by J. B. Keller (1958). The method is illustrated for the case of asymptotics of the solution of wave propagation from a point source in a plasma. The mathematical formulation of this problem is the following: \[ -{\partial^ 2u \over \partial t^ 2} + c^ 2 {\partial^ 2u \over \partial x^ 2}-\lambda^ 2b^ 2(x)u=\lambda \delta(x) r(t)e^{-i \lambda q(t)} =F(x,t,\lambda),u |_{t=0}=u_ t |_{t=0}=0; \tag{1} \] here \(x \in \mathbb{R}^ 3\), \(\lambda\) is the mean plasma frequency (which we use in the capacity of a large parameter; the corresponding dimensionless parameter is \(\lambda a/c\), where \(a\) is the characteristic length), and \(\lambda b(x)\) is the plasma frequency. \(b(x)>0\) is assumed.
The authors seek asymptotics which are uniform with respect to smoothness and the parameter \(\lambda\), \[ u(x,t,\lambda)=-\int \Phi (A_ 1,A_ 2,B,t,\tau)F(x,\tau,\lambda) d \tau. \]
35G10 Initial value problems for linear higher-order PDEs
78A45 Diffraction, scattering
78A05 Geometric optics