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Revista Colombiana de Matemáticas

versão impressa ISSN 0034-7426

Rev.colomb.mat. vol.50 no.1 Bogotá jan. 2016

https://doi.org/10.15446/recolma.v50n1.62200 

DOI: https://doi.org/10.15446/recolma.v50n1.62200

Bifurcation points of nonlinear operators: existence theorems, asymptotics and application to the Vlasov-Maxwell system

Puntos de bifurcación de los operadores no lineales: teoremas de existencia, asintótica y aplicación al sistema de Vlasov-Maxwell

Leonardo Rendón1, Alexandre V. Sinitsyn1, Nikolai A. Sidorov2

1 Universidad Nacional de Colombia, Bogotá, Colombia. lrendona@unal.edu.co, asinitsyne@unal.edu.co
2 Irkutsk State University, Irkutsk, Russia. sidorov@math.isu.runnet.ru


Abstract

Existence theorems about bifurcation points of solutions for nonlinear operator equation in Banach spaces are proved. The sufficient conditions of bifurcation of solutions of boundary-value problem for Vlasov-Maxwell system are obtained. The analytical method of Lyapunov-Schmidt-Trenogon is employed.

Keywords: plasma, bifurcation points, Conley index, nonlinear analysis, Vlasov-Maxwell system, Lyapunov-Schmidt-Trenogin method.


2010 Mathematics Subject Classification: 35J05, 35J10, 35K05.


Resumen

Se prueban teoremas de existencia de puntos de bifurcación en las soluciones de algunos operadores no lineales en espacios de Banach. Condiciones suficientes de bifurcación son obtenidas para las soluciones del problema de valor de frontera del sistema de Vlasov-Maxwell. Se emplea el método de Lyapunov-Schmidt-Trenogon.

Palabras claves: Plasma, bifurcación, índice de Conley, sistema Valsov-Maxwell, método de Lyapunov-Schmidt-Trenogin.


Texto completo disponible en PDF


References

[1] P. Braasch, Semilineare elliptische Differentialgleichungen und das Vlasov-Maxwell-System, P. Braasch, Dissertation, Herbert Utz Verlag Wissenschaft, Munchen, 1997.         [ Links ]

[2] C. C. Conley, Isolated invariant sets and the Morse index, CBMS. Regional Conf. Ser. Math. 38 (1978), AMS, Providence, R.I.         [ Links ]

[3] Y. Guo, Global weak solutions of the Vlasov-Maxwell system of plasma physics, Commun. Math. Phys. 154 (1993), 245-263.         [ Links ]

[4] ______, On steady states in a collisionless plasma, Comm. Pure Appl. Math. 11 (1996), 1145-1174.         [ Links ]

[5] H. Kielhöfer, A bifurcation theorem for potential operators, J. Funct. Anal. 77 (1988), 1-8.         [ Links ]

[6] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press., Oxford., 1964.         [ Links ]

[7] L. Kronecker, It Uber systeme von functionen mehrerer variables, Monats berichte de l'Academic ed Berlin. (1869), 159-198.         [ Links ]

[8] O. A. Ladyzhenskaya, Linear and Nonlinear Equations of Elliptic Type, Nauka, Moscow., 1964.         [ Links ]

[9] B. V. Loginov, Group Symmetry of the Lyapunov-Schmidt Branching Equation and Iterative Methods in the Problem of Bifurcation Points, Matematicheskii Sbornik. 182 (1991), no. 5, 681-690.         [ Links ]

[10] Y. A. Markov, Existence of stationary solutions of the Vlasov-Maxwell system Exact solutions, Mathematical Modelling. 1 (1989), 95-107.         [ Links ]

[11] ______, Steady-state solutions of the Vlasov-Maxwell system and their stability, Acta. Appl. Math. 28 (1992), 253-293.         [ Links ]

[12] G. Rein, Generic global solutions of the relativistic Vlasov-Maxwell system with nearly neutral innitial data, Commun. Math. Phys. 135 (1990), 41-78.         [ Links ]

[13] G. A. Rudykh, Stationary solutions of the system of Vlasov-Maxwell system, Doklady AN USSR. 33 (1988), 673-674.         [ Links ]

[14] ______, On Bifurcation Stationary Solutions of 2-Particle Vlasov-Maxwell System, Doklady Akademii Nauk SSSR. 304 (1989), no. 5, 1109-1112.         [ Links ]

[15] N. A. Sidorov, Investigation of points of bifurcation and continuous branches of solutions of the nonlinear equations, In Differential and Integral Equations, Irkutsk, Irkutsk University. (1972), 216-248.         [ Links ]

[16] ______, On bifurcating solutions of the nonlinear equations with potential branching equation, Doklady RAN. 23 (1981), 193-197.         [ Links ]

[17] ______, Points and surfaces of bifurcation of nonlinear operators with potential branching systems, Preprint. Irkutsk. Irkutsk Computing Center SB RAN. (1991).         [ Links ]

[18] ______, Explicite and Implicite Parametrization in the Construction of Branching Solutions, Matematicheskii Sbornik 186 (1995), no. 2, 129-140.         [ Links ]

[19] ______, On branching of solutions of the Vlasov-Maxwell system., Sibirsk. Matem. Zhyrnal. 37 (1996), 1367-1379.         [ Links ]

[20] ______, On nontrivial solutions and points of bifurcation of the Vlasov-Maxwell system, Doklady RAN. 349 (1996), 26-28.         [ Links ]

[21] ______, Analysis of bifurcation points and nontrivial branches of solutions to the stationary Vlasov-Maxwell system, Mat. Zametki. 62 (1997), no. 2, 268-292.         [ Links ]

[22] ______, Index theory in the bifurcation problem of solutions of the Vlasov-Maxwell system, Matem. Mod. 11 (1999), no. 9, 83-100.         [ Links ]

[23] ______, On bifurcation points of stationary Vlasov-Maxwell system with bifurcation directon, European consortium for mathematics in industry. Progress in industrial mathematics at ECMI-1998 Conference. Teubner, Stuttgart. (1999), 292-230.         [ Links ]

[24] ______, Interlaced branching equations in the theory of non-linear equations, Mat. Sb. 192 (2001), no. 7, 1070-1124.         [ Links ]

[25] ______, Bifurcation Points of Nolinear Equations, Nonlinear Analysis and Nonlinear Differential Equations. Edt. by V. A. Trenogin and A. T. Fillippov. M., 2003, Fizmatlit, 5-49.         [ Links ]

[26] ______, Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications. Series: Mathematics and Its Applications, Vol. 550, Springer Publ., 2003.         [ Links ]

[27] ______, Stationary Vlasov-Maxwell System in the Bounded Domain, Nonlinear Analysis and Nonlinear Differential Equations. Edt. by V. A. Trenogin and A. T. Fillippov. M., 2003, Fizmatlit, 50-84.         [ Links ]

[28] ______, Existence and construction of generalized solutions of nonlinear Volterra integral equations of the first kind, Differ. Equ. 42 (2006), 1312-1316.         [ Links ]

[29] ______, Solution of Volterra operator-integral equations in the nonregular case by the successive approximation method, Differ. Equ. 46 (2010), 882-891.         [ Links ]

[30] ______, Small solutions of nonlinear differential equations near branching points, Russ. Math. 55 (2011), no. 5, 43-50.         [ Links ]

[31] ______, On Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhood, Mathematical Notes. 91 (2012), no. 1, 90-104.         [ Links ]

[32] ______, Successive Approximations to Solutions of Nonlinear Equations with Vector Parameter in the Irregular Case, Sibirskii Zhurnal Industrial'noi Matematiki. 15 (2012), no. 1, 132-137.         [ Links ]

[33] V. A. Trenogin, Potentiality, group symmetry and bifurcation in the theory of branching equations, Different. and Integral Equat. 3 (1990), 145-154.         [ Links ]

[34] ______, Bifurcation, Potentiality, Group-Theoretical and Iterative Methods, Zeitschrift fur angewandte Mathematik und Mechanik. 76 (1996), 245-248.         [ Links ]

[35] V. V. Vedenyapin, Kinetic Boltzmann, Vlasov and Related Equations, Elsevier Publ., 2011.         [ Links ]

[36] M. M. Vainberg, Branching Theory of Solutions of Nonlinear Equations, Monographs and Textbooks on Pure and Applied Mathematics, Noordhoff International Publishing, Leyden., 1974.         [ Links ]

[37] A. A. Vlasov, Theory of Many-particles, U.S. Atomic Energy Commission, Technical Information Service Extension., 1950.         [ Links ]

(Recibido: septiembre de 2015 Aceptado: febrero de 2016)

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