Список выпусков > Серия «Математика». 2013. Том 4
Точки бифуркации нелинейных операторов: теоремы существования и приложения к исследованию систем Власова-Максвелла
Дан обзор теорем существования точек бифуркации решений нели-нейныхоператорных уравнений в банаховых пространствах. Получены достаточные условия ветвления решений граничныхзадач для систем Власова-Максвелла. При построении асимптотики решений граничной задачи используется аналитический метод Ляпунова-Шмидта-Треногина.
1. Braasch P. Semilineare elliptische Differentialgleichungen und das Vlasov-Maxwell-System / P. Braasch // Dissertation, Herbert Utz Verlag Wissenschaft. -Munchen, 1997.
2. Conley C. C. Isolated invariant sets and the Morse index / C .C. Conley // CBMS. Regional Conf. Ser. Math., 38, AMS, Providence, R. I. - 1978. - Vol. 38.
3. Guo Y. Global weak solutions of the Vlasov-Maxwell system of plasma physics / Y. Guo // Commun. Math. Phys. - 1993. - Vol. 154. - P. 245-263.
4. Guo Y. On steady states in a collisionless plasma / Y. Guo, C. G. Ragazzo // Comm. Pure Appl. Math. - 1996. - Vol. 11. - P. 1145-1174.
5. Kielhofer H. A bifurcation theorem for potential operators / H. Kielhofer // J. Funct. Anal. - 1988. - Vol. 77. - P. 1-8.
6. Krasnoselskii M. A. Topological Methods in the Theory of Nonlinear Integral Equations / M. A. Krasnoselskii. - Oxford : Pergamon Press., 1964.
7. Kronecker L. It Uber systeme von functionen mehrerer variables / L. Kronecker // Monats berichte de l'Academic ed Berlin. -1869. - P. 159-198.
8. Ladyzhenskaya O. A. Linear and Nonlinear Equations of Elliptic Type / O. V. Ladyzhenskaya, N. N. Uralzeva. - M. : Nauka, 1964.
9. Steady-state solutions of the Vlasov - Maxwell system and their stability / Y. A. Markov,G. A.Rudykh,N. A.Sidorov,A.V.Sinitsyn, D. A. Tolstonogov//Acta. Appl. Math. -1992. - Vol. 28. - P. 253-293.
10. Existence of stationary solutions of the Vlasov-Maxwell system Exact solutions /Y.A.Markov, G. A. Rudykh,N.A.Sidorov,A.V.Sinitsyn//Mathematical Modelling. - 1989. - Vol. 1. - P. 95-107.
11. Rein G. Generic global solutions of the relativistic Vlasov- Maxwell system with nearly neutral innitial data / G. Rein // Commun. Math. Phys. - 1990. - Vol. 135. - P. 41-78.
12. Rudykh G. A. Stationary solutions of the system of Vlasov-Maxwell system / G. A. Rudykh, N. A. Sidorov, A. V. Sinitsyn // Doklady AN USSR. - 1988. - Vol. 33. - P.673-674.
13. Sidorov N. A. On nontrivial solutions and points of bifurcation of the Vlasov-Maxwell system / N. A. Sidorov, A. V. Sinitsyn // Doklady RAN. - 1996. - Vol. 349. - P. 26-28.
14. Sidorov N. A. On branching of solutions of the Vlasov-Maxwell system. / N. A. Sidorov, A. V. Sinitsyn // Sibirsk. Matem. Zhyrnal. - 1996. - Vol. 37. - P. 1367-1379.
15. Sidorov N. A. On bifurcating solutions of the nonlinear equations with potential branching equation / N. A. Sidorov // Doklady RAN. - 1981. -Vol. 23. - P. 193-197.
16. Sidorov N. A. Points and surfaces of bifurcation of nonlinear operators with potential branching systems / N. A. Sidorov, V. A. Trenogin : Preprint. - Irkutsk : Irkutsk Computing Center SB RAN, 1991.
17. Sidorov N. A. Investigation of points of bifurcation and continuous branches of solutions of the nonlinear equations, // N. A. Sidorov, V. A. Trenogin // In Differential and Integral Equations. - Irkutsk : Irkutsk University, 1972. - P. 216-248.
18. Trenogin V. A. Potentiality, group symmetry and bifurcation in the theory of branching equations, / V. A. Trenogin, N. A. Sidorov, B. V. Loginov // Different. and Integral Equat. - 1990. - Vol. 3. - P. 145-154.
19. Vainberg M. M. Branching Theory of Solutions of Nonlinear Equations / M. M. Vainberg, V. A. Trenogin // Monographs and Textbooks on Pure and Applied Mathematics. - Leyden : Noordhoff International Publishing, 1974.
20. Vlasov A. A. Theory of Many-particles / A. A. Vlasov. - U.S. Atomic Energy Commission, Technical Information Service Extension, 1950.
21. Lyapunov - Schmidt Methods in Nonlinear Analysis and Applications / N. A. Sidorov, B. V. Loginov, A. V. Sinitsyn, M. V. Falaleev. - Springer Publ., 2003. -(Mathematics and Its Applications vol. 550).
22. Sidorov N. A. Bifurcation Points of Nolinear Equations / N. A. Sidorov, V. A. Trenogin // Nonlinear Analysis and Nonlinear Differential Equations / eds. by V. A. Trenogin and A. T. Fillippov. - M. : Fizmatlit, 2003. - P. 5-49.
23. Sidorov N. A. Stationary Vlasov-Maxwell System in the Bounded Domain / N. A. Sidorov, A. V. Sinitsyn // Nonlinear Analysis and Nonlinear Differential Equations / eds. by V. A. Trenogin and A. T. Fillippov. - M. : Fizmatlit, 2003. - P. 50-84.
24. Loginov B. V. Group Symmetry of the Lyapunov-Schmidt Branching Equation and Iterative Methods in the Problem of Bifurcation Points / B. V. Loginov, N. A. Sidorov // Matematicheskii Sbornik. - 1991. - Vol. 182 (5). - P. 681-690.
25. Sidorov N. A. Explicite and Implicite Parametrization in the Construction of Branching Solutions / N. A. Sidorov // Matematicheskii Sbornik. - 1995. - Vol. 186 (2). - P. 129-140.
26. Sidorov N. A. Interlaced branching equations in the theory of non-linear equations / N. A. Sidorov, V. R. Abdullin // Mat. Sb. - 2001. - Vol. 192(7). - P. 107—124.
27. Sidorov N. A. Analysis of bifurcation points and nontrivial branches of solutions to the stationary Vlasov -- Maxwell system / N. A. Sidorov, A. V. Sinitsyn // Mat. Zametki. - 1997. - Vol. 62(2). - P. 268-292.
28. Sidorov N. A. Index theory in the bifurcation problem of solutions of the Vlasov-Maxwell system / N. A. Sidorov, A. V. Sinitsyn // Matem. Mod. - 1999. - Vol. 11(9). - P. 83-100.
29. Sidorov N. A. On bifurcation points of stationary Vlasov-Maxwell system with bifurcation directon / N. A. Sidorov, A. V. Sinitsyn // European consortium for mathematics in industry. Progress in industrial mathematics at ECMI-1998Conference. - Teubner, Stuttgart, 1999. - P. 292-230.
30. Trenogin V. A. Bifurcation, Potentiality, Group-Theoretical and Iterative Methods / V. A. Trenogin, N. A. Sidorov, B. V. Loginov // Zeitschrift fur angewandte Mathematik und Mechanik. - 1996. - Vol. 76. - P. 245-248.
31. Rudykh G. A. On Bifurcation Stationary Solutions of 2-Particle Vlasov - Maxwell System / G. A. Rudykh, N. A. Sidorov, A. V. Sinitsyn // Doklady Akademii Nauk SSSR. - 1989. - Vol. 304 (5). - P. 1109-1112.
32. Sidorov N. A. Successive Approximations to Solutions of Nonlinear Equations with Vector Parameter in the Irregular Case / N. A. Sidorov, D. N. Sidorov, R. Yu. Leontiev // Sibirskii Zhurnal Industrial'noi Matematiki. - 2012. - Vol. 15(1). - P. 132-137.
33. Sidorov N. A. Existence and construction of generalized solutions of nonlinear Volterra integral equations of the first kind / N. A. Sidorov, D. N. Sidorov // Differ. Equ. - 2006. - Vol. 42. - P. 1312-1316.
34. Sidorov N. A. Solution of Volterra operator-integral equations in the nonregular case by the successive approximation method / N. A. Sidorov, D. N. Sidorov, A. V. Krasnik // Differ. Equ. - 2010. - Vol. 46. - P. 882-891.
35. Vedenyapin V. V. Kinetic Boltzmann, Vlasov and Related Equations / V. V. Vedenyapin, A. V. Sinitsyn, E. Dulov. - Elsevier Publ., 2011.
36. Sidorov N. A. On Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhood / N. A. Sidorov, R. Yu. Leont'iev, A. I. Dreglea // Mathematical Notes. - 2012. - Vol. 91(1). - P. 90-104.
37. Sidorov N. A. Small solutions of nonlinear differential equations near branching points / N. A. Sidorov, D. N. Sidorov // Russ. Math. - 2011. - Vol. 55 (5). P. 43-50.