A review of earlier generalizations of the classical Bateman solution, which involves an arbitrary function, is given. Further generalizations of it, getting the phase parametrized by m(m – 1) free real parameters, are built. Under a proper choice of the arbitrary function, such a solution may describe a Gaussian beam or a Gaussian packet. The approach is based upon a certain connection between solutions of a Srödinger-type equation and the wave equation. Bibliography: 37 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 393, 2011, pp. 167–177.
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Kiselev, A.P., Plachenov, A.B. Exact solutions of the m-dimensional wave equation from paraxial ones. Further generalizations of the Bateman solution. J Math Sci 185, 605–610 (2012). https://doi.org/10.1007/s10958-012-0944-7
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DOI: https://doi.org/10.1007/s10958-012-0944-7