Some Modern Aspects of Self-focusing Theory

Fibich, Gadi

doi:10.1007/978-0-387-34727-1_17

Cited by 13 publications

(13 citation statements)

References 60 publications

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“…In 2d, it appears as a model in nonlinear optics -see Fibich [7] for a review. When coupled with a nonlinear wave equation, it arises as the Zakharov system [32] in plasma physics.…”

Section: Then T < +∞ (The Solution Blows-up In Finite Time)mentioning

confidence: 99%

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

Holmer

Roudenko

2008

Commun. Math. Phys.

283

349

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Abstract. We consider the problem of identifying sharp criteria under which radial H 1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂ t u + ∆u + |u| 2 u = 0 scatter, i.e. approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the, where u 0 denotes the initial data, and M [u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution e it Q(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if, then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution e it Q(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if, then the solution blows-up in finite time. The technique employed is parallel to that employed by in their study of the energy-critical NLS.

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“…In 2d, it appears as a model in nonlinear optics -see Fibich [7] for a review. When coupled with a nonlinear wave equation, it arises as the Zakharov system [32] in plasma physics.…”

Section: Then T < +∞ (The Solution Blows-up In Finite Time)mentioning

confidence: 99%

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

Holmer

Roudenko

2008

Commun. Math. Phys.

283

349

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“…Most of the analyses on self focusing of the laser beams are devoted to the Gaussian beams [3][4][5][33][34][35][36][37][38][39][40][41][42][43]; nevertheless a few studies have also been published on the self focusing of super Gaussian beams [44][45][46], self trapping of degenerate modes of laser beams [47], and self trapping of Bessel beams [48] by considering the nature of different irradiance distributions of the beams. Apart from these investigations, recently the optical beams with central shadow, usually known as dark hollow beams (DHB) have received much attention from the physics community because of their wide and attractive applications in the field of modern optics, atomic optics and plasmas [49][50][51][52]; numerous experimental techniques [53][54][55] have also been developed for the production of the DHBs.…”

Section: Introductionmentioning

confidence: 99%

Focusing of Dark Hollow Gaussian Electromagnetic Beams in a Plasma With Relativistic-Ponderomotive Regime

MIshra¹,

Mishra²

2009

PIER B

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Abstract-This paper presents a theoretical model for the propagation of a hollow Gaussian electromagnetic beam [HGB], propagating in a plasma with dominant relativistic-ponderomotive nonlinearity. A paraxial like approach has been invoked to understand the nature of propagation; in this approach all the relevant parameters correspond to a narrow range around the irradiance maximum of the HGB. The critical curves for the propagation of various order HGBs have been discussed, and the dependence of the beam width parameter on distance of propagation has been evaluated for three typical cases viz. of steady divergence, oscillatory divergence and self focusing of the HGB.

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“…The values of H 0 for the three input powers are 0.1974x10 :% , 0.3948x10 :% and 0.1316x10 :b (in c.g.s). .Here e = β/α [34].The threshold power for collapse increases with input beam ellipticity and the threshold power for collapse is above the critical power (35).…”

Section: Resultsmentioning

confidence: 99%

A Study of Beam Parameters Using NLSE in Chalcogenide Glass Through Variational Method with a Gaussian Trial Function

Hazarika¹,

Das²,

Hazarika³

2015

AJOP

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Using variational method for an elliptical Gaussian optical beam trial function, self-action in bulk chalcogenide glass (Kerr media) is investigated. Emphasis is laid on the study of variation in beam width, curvature, phase and intensity of the beam with propagation distance. Solutions predict stationary self-focusing of the elliptical beam and an effective beam collapse at 10P cr input power. These study is significant in the choice of parameters in optical communications.

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Some Modern Aspects of Self-focusing Theory

Abstract: Summary. In this Chapter, we give a brief summary of the "present status" of self-focusing theory, while trying to highlight the fascinating evolution of this theory.

Cited by 13 publications

References 60 publications

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

Focusing of Dark Hollow Gaussian Electromagnetic Beams in a Plasma With Relativistic-Ponderomotive Regime

A Study of Beam Parameters Using NLSE in Chalcogenide Glass Through Variational Method with a Gaussian Trial Function

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