Geodesic
Reducing quasilinear systems to block triangular form
Sbornik. Mathematics, Tome 204 (2013) no. 3, pp. 438-462

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is concerned with systems of $n$ quasilinear partial differential equations of the first order with 2 independent variables. Using a geometric formalism for such equations, which goes back to Riemann, it is possible to assign a field of linear operators on an appropriate vector bundle to this type of quasilinear system. Several tests for a quasilinear system to be reducible to triangular or block triangular form are obtained in terms of this field; they supplement well known results on diagonalization and block diagonalization due to Haantjes and Bogoyavlenskij. Bibliography: 10 titles.
Keywords: block triangular quasilinear systems, block diagonal quasilinear systems, fields of linear operators, Nijenhuis tensors, Haantjes tensors. 
D. V. Tunitsky. Reducing quasilinear systems to block triangular form. Sbornik. Mathematics, Tome 204 (2013) no. 3, pp. 438-462. http://geodesic.mathdoc.fr/item/SM_2013_204_3_a5/
@article{SM_2013_204_3_a5,
     author = {D. V. Tunitsky},
     title = {Reducing quasilinear systems to block triangular form},
     journal = {Sbornik. Mathematics},
     pages = {438--462},
     year = {2013},
     volume = {204},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_3_a5/}
}
TY  - JOUR
AU  - D. V. Tunitsky
TI  - Reducing quasilinear systems to block triangular form
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 438
EP  - 462
VL  - 204
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_3_a5/
LA  - en
ID  - SM_2013_204_3_a5
ER  - 
%0 Journal Article
%A D. V. Tunitsky
%T Reducing quasilinear systems to block triangular form
%J Sbornik. Mathematics
%D 2013
%P 438-462
%V 204
%N 3
%U http://geodesic.mathdoc.fr/item/SM_2013_204_3_a5/
%G en
%F SM_2013_204_3_a5

[1] O. I. Bogoyavlenskij, “Block-diagonalizability problem for hydrodynamic type systems”, J. Math. Phys., 47:6 (2006), 063502 | DOI | MR | Zbl

[2] R. Courant, D. Hilbert, Methods of mathematical physics, v. 2, Partial differential equations, Wiley, New York, 1989 | MR | Zbl | Zbl

[3] J. Haantjes, “On $X_m$-forming sets of eigenvectors”, Nederl. Akad. Wetensch. Proc. Ser. A., 58 (1955), 158–162 | MR | Zbl

[4] M. Postnikov, Lecons de géométrie. Géométrie différentielle, Traduit Russe Math., Mir, Moscow, 1990 | MR | MR | Zbl | Zbl

[5] M. Postnikov, Lectures in geometry. Semester II: Linear algebra and differential geometry, Mir, Moscow, 1982 | MR | Zbl

[6] A. Nijenhuis, “$X_{n-1}$-forming sets of eigenvectors”, Nederl. Akad. Wetensch. Proc. Ser. A., 54 (1951), 200–212 | MR | Zbl

[7] Sh. Kobayashi, K. Nomizu, Foundations of differential geometry, v. 1, Interscience Publ., New York–London–Sydney, 1963 | MR | MR | Zbl | Zbl

[8] O. I. Bogoyavlenskij, “Algebraic identities for the Nijenhuis tensors”, Differential Geom. Appl., 24:5 (2006), 447–457 | DOI | MR | Zbl

[9] B. L. Rozhdestvenskij, N. N. Yanenko, Systems of quasilinear equations and their applications to gas dynamics, Transl. Math. Monogr., 55, Amer. Math. Soc., Providence, RI, 1983 | MR | MR | Zbl | Zbl

[10] F. W. Warner, Foundations of differentiable manifolds and Lie groups, Foresman, Glenview, IL–London, 1971 | MR | MR | Zbl | Zbl