Alterman, Deborah; Rauch, Jeffrey The linear diffractive pulse equation. (English) Zbl 1001.35015 Methods Appl. Anal. 7, No. 2, 263-274 (2000). Summary: The asymptotic description of short wavelength wavetrain solutions of constant coefficient linear hyperbolic partial differential equations leads, for times of order 1, to amplitudes which satisfy the linear transport equation along rays. For linear phases and times of order \(1/\varepsilon\) with wavelength \(\approx O(\varepsilon)\), diffractive effects become important and the amplitudes satisfy a linear Schrödinger equation. The asymptotic description of pulse solutions involves the same linear transport equation for time of order one. For times of order \(1/\varepsilon\), instead of a Schrödinger equation, the amplitudes satisfy a partial differential equation we call the Linear Diffractive Pulse Equation (LDPE). Nonlinear analogues of all these results are known for solutions of the appropriate amplitudes. In this paper we examine in some detail the initial value problem for the LDPE. Cited in 4 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35G10 Initial value problems for linear higher-order PDEs 35Q60 PDEs in connection with optics and electromagnetic theory Keywords:linear Schrödinger equation; short wavelength wavetrain solutions PDFBibTeX XMLCite \textit{D. Alterman} and \textit{J. Rauch}, Methods Appl. Anal. 7, No. 2, 263--274 (2000; Zbl 1001.35015) Full Text: DOI