Geodesic
Laplace transformations of hydrodynamic-type systems in Riemann invariants
Teoretičeskaâ i matematičeskaâ fizika, Tome 110 (1997) no. 1, pp. 86-97 
E. V. Ferapontov. Laplace transformations of hydrodynamic-type systems in Riemann invariants. Teoretičeskaâ i matematičeskaâ fizika, Tome 110 (1997) no. 1, pp. 86-97. http://geodesic.mathdoc.fr/item/TMF_1997_110_1_a6/
@article{TMF_1997_110_1_a6,
     author = {E. V. Ferapontov},
     title = {Laplace transformations of hydrodynamic-type systems in {Riemann} invariants},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {86--97},
     year = {1997},
     volume = {110},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1997_110_1_a6/}
}
TY  - JOUR
AU  - E. V. Ferapontov
TI  - Laplace transformations of hydrodynamic-type systems in Riemann invariants
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1997
SP  - 86
EP  - 97
VL  - 110
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_1997_110_1_a6/
LA  - ru
ID  - TMF_1997_110_1_a6
ER  - 
%0 Journal Article
%A E. V. Ferapontov
%T Laplace transformations of hydrodynamic-type systems in Riemann invariants
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1997
%P 86-97
%V 110
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_1997_110_1_a6/
%G ru
%F TMF_1997_110_1_a6

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

The conserved densities of hydrodynamic-type systems in Riemann invariants satisfy a system of linear second-order partial differential equations. For linear systems of this type, Darboux introduced Laplace transformations, which generalize the classical transformations of a second-order scalar equation. It is demonstrated that the Laplace transformations can be pulled back to transformations of the corresponding hydrodynamic-type systems. We discuss finite families of hydrodynamic-type systems that are closed under the entire set of Laplace transformations. For $3\times3$ systems in Riemann invariants, a complete description of closed quadruples is proposed. These quadruples appear to be related to a special quadratic reduction of the $(2+1)$-dimensional 3-wave system.

[1] S. P. Tsarev, Izv. AN SSSR, ser. matem., 54:5 (1990), 1048–1068 | MR | Zbl

[2] G. Darboux, Lecons sur la theorie generale des surfaces, Part 4, Gautier-Villars, Paris

[3] A. B. Shabat, R. A. Yamilov, “To transformation theory of two-dimensional integrable systems”, Phys. Lett. A (to appear) | MR

[4] R. I. Yamilov, “Classification of Toda type scalar lattices”, Proc. 8th Int. Workshop on Nonlinear Evolution Equations and Dynamical Systems, World Sci. Publishing, 1993, 423–431 | MR

[5] A. B. Shabat, R. I. Yamilov, Algebra i analiz, 2:2 (1990), 183–208 | MR

[6] A. P. Veselov, S. P. Novikov, UMN, 1995, no. 6, 171–172 | MR | Zbl

[7] T. L. Kozmina, DAN, 55:3 (1947), 187–189 | Zbl

[8] R. V. Smirnov, DAN, 71:3 (1950), 437–439 | MR | Zbl

[9] T. A. Shulman, DAN, 85:3 (1952), 501–504 | MR

[10] M. A. Akivis, V. V. Goldberg, Rend. Sem. Mat. Messina, ser. II, 1 (1991), 9–29 | MR | Zbl

[11] N. Kamran, K. Tenenblat, Duke Math. J., 84:1 (1996), 237–266 | DOI | MR | Zbl

[12] C. Athorne, Phys. Lett. A, 206 (1995), 162–166 | DOI | MR | Zbl