Periodic solutions of the Brillouin electron beam focusing equation

Maurizio Garrione; Manuel Zamora

doi:10.3934/cpaa.2014.13.961

Article Contents

2014, Volume 13, Issue 2: 961-975. Doi: 10.3934/cpaa.2014.13.961

This issue Previous Article Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space Next Article

Periodic solutions of the Brillouin electron beam focusing equation

Maurizio Garrione^1, and
Manuel Zamora^2,

1.
Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 1, 60131, Ancona
2.
Departamento de Matemática Aplicada, Universidad de Granada, 18071, Granada

Received: January 31, 2013

Revised: August 31, 2013

Published: February 2014

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Abstract

Quite unexpectedly with respect to the numerical and analytical results found in literature, we establish a new range for the real parameter $b$ for which the existence of $2\pi-$periodic solutions of the Brillouin focusing beam equation \begin{eqnarray} \ddot{x}+b(1+\cos t)x=\frac{1}{x} \end{eqnarray} is guaranteed. This is possible thanks to suitable nonresonance conditions acting on the rotation number of the solutions in the phase plane.

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Mathematics Subject Classification: Primary: 34B15, 34C25; Secondary: 34B16.

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