Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems

Mihai Bostan; Eric Sonnendrücker

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 40, Issue: 6, page 1023-1052
  • ISSN: 0764-583X

Abstract

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We study the existence of spatial periodic solutions for nonlinear elliptic equations - Δ u + g ( x , u ( x ) ) = 0 , x ∈ ℝ N where g is a continuous function, nondecreasing w.r.t. u. We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions g are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations.

How to cite

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Bostan, Mihai, and Sonnendrücker, Eric. "Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems." ESAIM: Mathematical Modelling and Numerical Analysis 40.6 (2007): 1023-1052. <http://eudml.org/doc/194343>.

@article{Bostan2007,
abstract = { We study the existence of spatial periodic solutions for nonlinear elliptic equations $- \Delta u \, + \, g(x,u(x)) = 0, \;x \in \{\mathbb R\}^N$ where g is a continuous function, nondecreasing w.r.t. u. We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions g are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations. },
author = {Bostan, Mihai, Sonnendrücker, Eric},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear elliptic equations; periodic solutions; existence and uniqueness; electron beam focusing system.; nonlinear elliptic equations; existence and uniqueness; electron beam focusing system},
language = {eng},
month = {2},
number = {6},
pages = {1023-1052},
publisher = {EDP Sciences},
title = {Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems},
url = {http://eudml.org/doc/194343},
volume = {40},
year = {2007},
}

TY - JOUR
AU - Bostan, Mihai
AU - Sonnendrücker, Eric
TI - Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/2//
PB - EDP Sciences
VL - 40
IS - 6
SP - 1023
EP - 1052
AB - We study the existence of spatial periodic solutions for nonlinear elliptic equations $- \Delta u \, + \, g(x,u(x)) = 0, \;x \in {\mathbb R}^N$ where g is a continuous function, nondecreasing w.r.t. u. We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions g are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations.
LA - eng
KW - Nonlinear elliptic equations; periodic solutions; existence and uniqueness; electron beam focusing system.; nonlinear elliptic equations; existence and uniqueness; electron beam focusing system
UR - http://eudml.org/doc/194343
ER -

References

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  11. Z. Meiyue, C. Taiyoung, L. Wenbin and J. Yong, Existence of positive periodic solution for the electron beam focusing system. Math. Meth. Appl. Sci.28 (2005) 779–788.  
  12. A. Nouri, Paraxial approximation of the Vlasov-Maxwell system: laminar beams. Math. Models Meth. Appl. Sci.4 (1994) 203–221.  
  13. P.-A. Raviart, Paraxial approximation of the stationary Vlasov-Maxwell equations, Nonlinear partial differential equations and their applications. Collège de France Seminar, vol. XIII Paris (1991–1993), Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow 302 (1994) 158–171.  
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