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

Published online by Cambridge University Press:  15 February 2007

Mihai Bostan  and
Eric Sonnendrücker
Mihai Bostan
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France. [email protected]
Eric Sonnendrücker
Affiliation:
IRMA, Université Louis Pasteur, rue René Descartes, 67084 Strasbourg Cedex, France. [email protected]
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Abstract

We study the existence of spatial periodic solutions for nonlinearelliptic equations $- \Delta u \, + \, g(x,u(x)) = 0, \;x \in {\mathbb R}^N$ where g is a continuous function, nondecreasing w.r.t. u. Wegive necessary and sufficient conditions for the existence ofperiodic solutions. Some cases with nonincreasing functions gare investigated as well. As an application we analyze themathematical model of electron beam focusing system and we provethe existence of positive periodic solutions for the envelopeequation. We present also numerical simulations.

Keywords

Nonlinear elliptic equationsperiodic solutionsexistence anduniquenesselectron beam focusing system.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 6 , November 2006 , pp. 1023 - 1052
Copyright
© EDP Sciences, SMAI, 2007

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References

M. Bostan, Solutions périodiques des équations d'évolution. C. R. Acad. Sci., Ser. I, Math. 332 (2001) 401–404.
M. Bostan, Periodic solutions for evolution equations. Electron. J. Diff. Eqns., Monograph 3 (2002) 41.
Brezis, H., Problèmes unilatéraux. J. Math. Pures Appl. 51 (1972) 1–64.
R.C. Davidson and H. Qin, Physics of charged particle beams in high energy accelerators. Imperial College Press, World Scientific Singapore (2001).
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.
F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation. Research report INRIA, No. 5547 (2004).
I.M. Kapchinsky and V.V. Vladimirsky, Proceedings of the 9th international conference on high energy accelerators, CERN Geneva (1959) 274.
D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications. Academic Press, New York, London (1980).
Laval, G., Mas-Gallic, S. and Raviart, P.-A., Paraxial approximation of ultra-relativistic intense beams. Numer. Math. 1 (1994) 33–60. CrossRef
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non-linéaires. Dunod Gauthier-Villars (1969).
Meiyue, Z., Taiyoung, C., Wenbin, L. and Yong, J., Existence of positive periodic solution for the electron beam focusing system. Math. Meth. Appl. Sci. 28 (2005) 779–788.
Nouri, A., Paraxial approximation of the Vlasov-Maxwell system: laminar beams. Math. Models Meth. Appl. Sci. 4 (1994) 203–221. CrossRef
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.
M. Reiser, Theory and design of charged-particle beams. Wiley, New York (1994).