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

top
We study the existence of spatial periodic solutions for nonlinear elliptic equations - Δ u + g ( x , u ( x ) ) = 0 , x 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

top

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

top
  1. M. Bostan, Solutions périodiques des équations d'évolution. C. R. Acad. Sci., Ser. I, Math.332 (2001) 401–404.  Zbl1047.34065
  2. M. Bostan, Periodic solutions for evolution equations. Electron. J. Diff. Eqns., Monograph3 (2002) 41.  Zbl1010.34060
  3. H. Brezis, Problèmes unilatéraux. J. Math. Pures Appl.51 (1972) 1–64.  Zbl0237.35001
  4. R.C. Davidson and H. Qin, Physics of charged particle beams in high energy accelerators. Imperial College Press, World Scientific Singapore (2001).  
  5. P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Models Meth. Appl. Sci.3 (1993) 513–562.  Zbl0787.35110
  6. F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation. Research report INRIA, No. 5547 (2004).  Zbl1109.78013
  7. I.M. Kapchinsky and V.V. Vladimirsky, Proceedings of the 9th international conference on high energy accelerators, CERN Geneva (1959) 274.  
  8. D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications. Academic Press, New York, London (1980).  Zbl0457.35001
  9. G. Laval, S. Mas-Gallic and P.-A. Raviart, Paraxial approximation of ultra-relativistic intense beams. Numer. Math.1 (1994) 33–60.  Zbl0816.65119
  10. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non-linéaires. Dunod Gauthier-Villars (1969).  Zbl0189.40603
  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.  Zbl1069.34065
  12. A. Nouri, Paraxial approximation of the Vlasov-Maxwell system: laminar beams. Math. Models Meth. Appl. Sci.4 (1994) 203–221.  Zbl0803.35148
  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.  
  14. M. Reiser, Theory and design of charged-particle beams. Wiley, New York (1994).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.