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Relatively distortion-free waves for the \(m\)-dimensional wave equation. (English. Russian original) Zbl 1028.35038

Differ. Equ. 38, No. 8, 1206-1207 (2002); translation from Differ. Uravn. 38, No. 8, 1128-1129 (2002).
From the text: Relatively distortion-free waves are defined as solutions of the form \[ u= gf(\theta)\tag{1} \] of hyperbolic equations, where the phase \(\theta\) and the distortion factor \(g\) are given functions of the space variables and time and the function \(f(\theta)\) describing the wave shape is arbitrary. We are interested in simple explicit solutions of the form (1) of the wave equation \[ \square u= 0,\quad \square= {\partial^2\over\partial x^2_1}+ {\partial^2\over\partial x^2_2}+\cdots+ {\partial^2\over\partial x^2_m}- {1\over c^2} {\partial^2\over\partial t^2},\tag{2} \] \(c=\text{const}\), for an arbitrary \(m> 1\).
Theorem. Let \[ \theta= x_m- ct+ \sum^{m-1}_{j=1} x^2_j(\beta- E_j)^{-1}, \] where \(\beta= x_m+ ct\) and the \(E_j\), \(j= 1,\dots, m-1\), are arbitrary complex constants. Then the expression (1) with \[ g= \prod^{m-1}_{j=1} (\beta- E_j)^{-1/2}, \] when the branches of square roots can be chosen arbitrarily, is a solution (2) for an arbitrary function \(f(\theta)\).

MSC:

35C05 Solutions to PDEs in closed form
35L05 Wave equation
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