Interlaced branching equations in the theory of non-linear equations

N. A. Sidorov   ; V. R. Abdullin

Sbornik. Mathematics, Tome 192 (2001) no. 7, pp. 1035-1052

Résumé

Necessary conditions for inheriting the interlacing property of a non-linear equation by the branching system are obtained. The case when the pair of linear operators interlacing the equation consists of projections or parametric families of linear operators is considered. New conditions are presented which allow one to reduce the number of the equations in the branching system and extend the range of applications of the method of successive approximations in the branching theory of non-linear equations. Solutions depending on free parameters belonging to certain hypersurfaces in Euclidean spaces are considered. The results obtained add to and extend earlier results on the applications of group analysis in branching theory.

Bibliographie

[1] Vainberg M. M., Trenogin V. A., Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR

[2] Yudovich V. I., “Svobodnaya konvektsiya i vetvlenie”, PMM, 31:1 (1969), 101–111 | MR

[3] Loginov B. V., Trenogin V. A., “Ob ispolzovanii gruppovykh svoistv dlya opredeleniya mnogoparametricheskikh semeistv reshenii nelineinykh uravnenii”, Matem. sb., 85 (127) (1971), 440–454 | MR | Zbl

[4] Loginov B. V., Trenogin V. A., “Ob ispolzovanii gruppovoi invariantnosti v teorii vetvleniya”, Differents. uravneniya, 11 (1975), 1709–1712 | MR | Zbl

[5] Loginov B. V., Teoriya vetvleniya reshenii nelineinykh uravnenii v usloviyakh gruppovoi invariantnosti, Izd-vo FAN, Tashkent, 1985 | MR

[6] Loginov B. V., “Ob invariantnykh resheniyakh v teorii vetvleniya”, Dokl. AN SSSR, 246:5 (1979), 1048–1051 | MR | Zbl

[7] Loginov B. V., “Vetvlenie reshenii nelineinykh uravnenii i gruppovaya simmetriya”, Vestn. SamGU, 1998, no. 4 (10), 15–70 | MR | Zbl

[8] Ovsyanikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR

[9] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983 | MR

[10] Sattinger D. H., Group theoretical methods in bifurcation theory, Lecture Notes in Math., 762, Springer-Verlag, Berlin, 1979 | MR | Zbl

[11] Vanderbauwhede A., Local bifurcation and symmetry, Res. Notes Math., 75, Pitman, Boston, 1982 | MR | Zbl

[12] Golubitsky M., Keyfitz B. L., Schaeffer D. G., “A singularity theory analysis of a thermal-chainbranching model for the explosion peninsula”, Comm. Pure Appl. Math., 34 (1981), 433–463 | DOI | MR | Zbl

[13] Stewart J., “Applications of catastrophe theory to the physical sciences”, Phys. D, 2 (1981), 245–305 | DOI | MR

[14] Chillingworth D. R. J., “Notes on some recent methods in bifurcation theory”, Banach Center Publ., 15, 1985, 161–174 | MR | Zbl

[15] Loginov B. V., “Determination of the branching equation by its group symmetry – Andronov–Hopf bifurcation”, Nonlinear Anal., 28:12 (1997), 2033–2047 | DOI | MR | Zbl

[16] Keller H. B., “Numerical solution of bifurcation and nonlinear eigenvalue problems”, Application of bifurcation theory, ed. P. Rabinowitz, Academic Press, New York, 1977, 359–384 | MR

[17] Sidorov N. A., Sinitsin A. V., “Issledovanie tochek bifurkatsii i netrivialnykh vetvei reshenii sistemy Vlasova–Maksvella”, Matem. zametki, 62:2 (1997), 268–292 | MR | Zbl

[18] Vorovich I. I., Nonlinear theory of shallow shells, Springer-Verlag, New York, 1999 | MR | Zbl

[19] Grigolyuk E. I., Shalashilin V. I., Problemy nelineinogo differentsirovaniya: metod prodolzheniya po parametru v nelineinykh zadachakh mekhaniki tverdogo tela, Nauka, M., 1988 | MR | Zbl

[20] Buchanan M., Dorning J., “Nonlinear waves in collisionless plasmas”, Phys. Rev. E, 52:2 (1995), 3015–3033 | DOI | MR

[21] Loginov B. V., Sidorov N. A., “Gruppovaya simmetriya uravneniya razvetvleniya Lyapunova–Shmidta i iteratsionnye metody v zadache o tochke bifurkatsii”, Matem. sb., 182:5 (1991), 681–691 | MR | Zbl

[22] Sidorov N. A., “Yavnaya i neyavnaya parametrizatsiya pri postroenii razvetvlyayuschikhsya reshenii iteratsionnymi metodami”, Matem. sb., 186:2 (1995), 129–141 | MR | Zbl

[23] Sidorov N. A., “$N$-stupenchatyi iteratsionnyi metod v teorii vetvleniya reshenii nelineinykh uravnenii”, Sib. matem. zhurn., 38:2 (1997), 383–395 | MR | Zbl

[24] Sidorov N. A., Abdullin V. R., “Interlaced branching equations and invariance in the theory of nonlinear equations”, Symmetry and perturbation theory, International workshop SPT'98 (Roma), World Scientific, Singapore, 1999 | MR | Zbl

[25] Sidorov N. A., Trenogin V. A., “Usloviya potentsialnosti uravneniya razvetvleniya i tochki bifurkatsii nelineinykh operatorov”, Uzb. matem. zhurn., 1992, no. 2, 40–49 | MR

[26] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980 | MR | Zbl

[27] Sidorov N. A., Abdullin V. R., Spletayuschie uravneniya razvetvleniya v teorii nelineinykh uravnenii, Preprint No 1. Ser. ANN IRO, Izd-vo Irkutskogo un-ta, Irkutsk, 1999

[28] Sidorov N. A., Sinitsyn A. V., “On bifurcation points of the stationary Vlasov–Maxwell system with bifurcation direction”, Progress in industrial mathematics at ESMI 98, eds. L. Arkeryd, J. Bergh, P. Brenner, R. Pettersson, B. G. Teubner, Stuttgart, 1999, 295–303 | MR

[29] Sidorov N. A., Romanova O. A., Difference-differential equations with the Fredholm operator in the main part, Report TW 243, Katholike Universiteit Leuven, Department of computer science, Belgium, 1996