### 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 language | English (US) |
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Pages (from-to) | 595-610 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 143 |

Issue number | 2 |

DOIs | |

State | Published - 2015 |

### Fingerprint

### 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

**Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation.** / Mahalov, Alex; Suslov, Sergei.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation

AU - Mahalov, Alex

AU - Suslov, Sergei

PY - 2015

Y1 - 2015

N2 - 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.

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.

KW - Airy–Gaussian–Hermite beams

KW - Complex Ginsburg–Landau equations

KW - Gaussian–Hermite beams

KW - Green’s function

KW - Nonlinear Schrödinger equations

KW - Paraxial wave equations

UR - http://www.scopus.com/inward/record.url?scp=84948382869&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84948382869&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2014-12295-7

DO - 10.1090/S0002-9939-2014-12295-7

M3 - Article

AN - SCOPUS:84948382869

VL - 143

SP - 595

EP - 610

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -