Geodesic
$F$-manifolds and integrable systems of hydrodynamic type
Archivum mathematicum, Tome 47 (2011) no. 3, pp. 163-180 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We investigate the role of Hertling-Manin condition on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of $F$-manifold with compatible connection generalizing a structure introduced by Manin.
We investigate the role of Hertling-Manin condition on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of $F$-manifold with compatible connection generalizing a structure introduced by Manin.
Classification : 35Q35, 53B05, 53D45
Keywords: F-manifolds; Frobenius manifolds; integrable systems; PDEs of hydrodynamic type 
@article{ARM_2011_47_3_a0,
     author = {Lorenzoni, Paolo and Pedroni, Marco and Raimondo, Andrea},
     title = {$F$-manifolds and integrable systems of hydrodynamic type},
     journal = {Archivum mathematicum},
     pages = {163--180},
     year = {2011},
     volume = {47},
     number = {3},
     mrnumber = {2852379},
     zbl = {1249.35267},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2011_47_3_a0/}
}
TY  - JOUR
AU  - Lorenzoni, Paolo
AU  - Pedroni, Marco
AU  - Raimondo, Andrea
TI  - $F$-manifolds and integrable systems of hydrodynamic type
JO  - Archivum mathematicum
PY  - 2011
SP  - 163
EP  - 180
VL  - 47
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/ARM_2011_47_3_a0/
LA  - en
ID  - ARM_2011_47_3_a0
ER  - 
%0 Journal Article
%A Lorenzoni, Paolo
%A Pedroni, Marco
%A Raimondo, Andrea
%T $F$-manifolds and integrable systems of hydrodynamic type
%J Archivum mathematicum
%D 2011
%P 163-180
%V 47
%N 3
%U http://geodesic.mathdoc.fr/item/ARM_2011_47_3_a0/
%G en
%F ARM_2011_47_3_a0
Lorenzoni, Paolo; Pedroni, Marco; Raimondo, Andrea. $F$-manifolds and integrable systems of hydrodynamic type. Archivum mathematicum, Tome 47 (2011) no. 3, pp. 163-180. http://geodesic.mathdoc.fr/item/ARM_2011_47_3_a0/

[1] Benney, D. J.: Some properties of long nonlinear waves. Stud. Appl. Math. 52 (1973), 45–50. | Zbl

[2] Chang, Jen-Hsu: On the waterbag model of the dispersionless KP hierarchy (II). J. Phys. A: Math. Theor. 40 (2007), 12973–12785. | DOI | MR | Zbl

[3] Dubrovin, B. A.: Geometry of 2D topological field theories. Integrable Systems and Quantum Groups (Francaviglia, M., Greco, S., eds.), vol. 1620, Montecatini Terme, 1993, springer lecture notes in math. ed., 1996, pp. 120–348. | MR | Zbl

[4] Dubrovin, B. A.: Flat pencils of metrics and Frobenius manifolds. Integrable systems and algebraic geometry, World Sci. Publ., River Edge, NJ, 1998, pp. 47–72. | MR | Zbl

[5] Dubrovin, B. A., Liu, S. Q., Zhang, Y.: Frobenius manifolds and central invariants for the Drinfeld–Sokolov biHamiltonian structures. Adv. Math. 120 (3) (2008), 780–837. | DOI | MR | Zbl

[6] Dubrovin, B. A., Novikov, S. P.: Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. Uspekhi Mat. Nauk 44 (1989), 35–124, English translation in Russ. Math. Surveys 44 (1989), 35–124. | MR | Zbl

[7] Dubrovin, B.A.: Differential geometry of strongly integrable systems of hydrodynamic type. Funct. Anal. Appl. 24 (4) (1990), 280–285, (English. Russian original), translation from Funkts. Anal. Prilozh. 24, No.4, 25-30 (1990). | DOI | MR | Zbl

[8] Ferapontov, E. V.: Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type. Funct. Anal. Appl. 25 (3) (1991), 195–204. | DOI | MR | Zbl

[9] Ferguson, J. T., Strachan, I. A. B.: Logarithmic deformations of the rational superpotential/Landau-Ginzburg construction of solutions of the WDVV equations. Comm. Math. Phys. 280 (2008), 1–25. | DOI | MR | Zbl

[10] Gibbons, J., Lorenzoni, P., Raimondo, A.: Purely nonlocal Hamiltonian formalism for systems of hydrodynamic type. arXiv:0812.3317. | MR | Zbl

[11] Gibbons, J., Lorenzoni, P., Raimondo, A.: Hamiltonian structure of reductions of the Benney system. Comm. Math. Phys. 287 (2009), 291–322. | DOI | MR

[12] Gibbons, J., Tsarev, S. P.: Reductions of the Benney equations. Phys. Lett. A 211 (1) (1996), 19–24. | DOI | MR | Zbl

[13] Gibbons, J., Tsarev, S. P.: Conformal maps and reductions of the Benney equations. Phys. Lett. A 258 (4–6) (1999), 263–271. | DOI | MR | Zbl

[14] Hertling, C.: Multiplication on the tangent bundle. arXiv:math/9910116.

[15] Hertling, C., Manin, Y.: Weak Frobenius manifolds. Internat. Math. Res. Notices 6 (1999), 277–286. | DOI | MR | Zbl

[16] Konopelchenko, B. G., Magri, F.: Coisotropic deformations of associative algebras and dispersionless integrable hierarchies. Comm. Math. Phys. 274 (2007), 627–658. | DOI | MR | Zbl

[17] Lebedev, D., Manin, Y.: Conservation laws and Lax representation of Benney’s long wave equations. Phys. Lett. A 74 (1979), 154–156. | DOI | MR

[18] Manin, Y.: $F$-manifolds with flat structure and Dubrovin’s duality. Adv. Math. 198 (1) (2005), 5–26. | DOI | MR | Zbl

[19] Mokhov, O. I.: Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and associativity equations. Funct. Anal. Appl. 40 (2006), 11–23. | DOI | MR

[20] Mokhov, O. I.: Frobenius manifolds as a special class of submanifolds in pseudo-Euclidean spaces. Geometry, Topology, and Mathematical Physics, vol. 224, Amer. Math. Soc. Transl. Ser. 2, 2008, pp. 213–246. | MR | Zbl

[21] Mokhov, O. I., Ferapontov, E. V.: Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature. Russ. Math. Surv. 45 (3) (1990), 218–219. | DOI | MR

[22] Pavlov, M. V.: Integrability of Egorov systems of hydrodynamic type. Teoret. Mat. Fiz. 150 (2) (2007), 263–285, (Russian) translation in Theoret. and Math. Phys. 150 (2) (2007), 225–243. | MR

[23] Pavlov, M. V., Svinolupov, S. I., Sharipov, R. A.: An invariant criterion for hydrodynamic integrability. Funktsional. Anal. i Prilozhen 30 (1) (1996), 18–29, 96, (Russian) translation in Funct. Anal. Appl. 30 (1) (1996), 15–22. | DOI | MR

[24] Pavlov, M. V., Tsarev, S. P.: Tri-Hamiltonian structures of Egorov systems of hydrodynamic type. Funktsional. Anal. i Prilozhen. 37 (1) (2003), 38–54, (Russian). | DOI | MR | Zbl

[25] Strachan, I. A. B.: Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures. Differential Geom. Appl. 20 (1) (2004), 67–99. | DOI | MR | Zbl

[26] Tsarev, S. P.: The geometry of Hamiltonian systems of hydrodynamic type. The generalised hodograph transform. USSR Izv. 37 (1991), 397–419. | DOI | MR

[27] Zakharov, V. E.: Benney equations and quasiclassical approximation in the inverse problem. Funktional. Anal. i Prilozhen 14 (1980), 15–24. | MR