Abstract

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
Volume143
Issue number2
DOIs
StatePublished - 2015

Fingerprint

Quadratic Approximation
Inhomogeneous Media
Linear Approximation
Wave equations
Wave equation
Particle beams
Explicit Solution
Nonlinear equations
Laser Beam
Parabolic Equation
Laser beams
Waveguide
Thin Films
Nonlinear Equations
Waveguides
Thin films
Review
Form

Keywords

  • 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

Cite this

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title = "Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation",
abstract = "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.",
keywords = "Airy–Gaussian–Hermite beams, Complex Ginsburg–Landau equations, Gaussian–Hermite beams, Green’s function, Nonlinear Schr{\"o}dinger equations, Paraxial wave equations",
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AB - 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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KW - Complex Ginsburg–Landau equations

KW - Gaussian–Hermite beams

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KW - Paraxial wave equations

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