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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation
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by Alex Mahalov and Sergei K. Suslov PDF
Proc. Amer. Math. Soc. 143 (2015), 595-610 Request permission


We construct explicit solutions of the inhomogeneous parabolic wave equation in a linear and quadratic approximation. As examples, oscillating laser beams in a $1D$ parabolic waveguide, spiral light beams in $2D$ varying media and an effect of superfocusing of particle beams in a thin monocrystal film are briefly discussed. Transformations of nonlinear equations into the corresponding autonomous and homogeneous forms are found and a review of important applications is also given.
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Additional Information
  • Alex Mahalov
  • Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287–1804
  • Email:
  • Sergei K. Suslov
  • Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287–1804
  • Email:
  • Received by editor(s): March 26, 2013
  • Published electronically: October 28, 2014
  • Additional Notes: This research was partially supported by AFOSR grant FA9550-11-1-0220.
  • Communicated by: Ken Ono
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 595-610
  • MSC (2010): Primary 35Q55, 35Q51; Secondary 35C05, 81Q05
  • DOI:
  • MathSciNet review: 3283647