5 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)595-610
Number of pages16
JournalProceedings of the American Mathematical Society
Issue number2
StatePublished - 2015


  • Airy–Gaussian–Hermite beams
  • Complex Ginsburg–Landau equations
  • Gaussian–Hermite beams
  • Green’s function
  • Nonlinear Schrödinger equations
  • Paraxial wave equations

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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