×

zbMATH — the first resource for mathematics

Deformations of semisimple bihamiltonian structures of hydrodynamic type. (English) Zbl 1079.37058
The authors study the problem of classification of deformations of a given bi-Hamiltonian structure of hydrodynamic type, these deformations depend on a parameter \(\varepsilon\) which is called the dispersion parameter. The deformed bi-Hamiltonian structure has the form \[ \begin{split}\{u^i(x), u^j(y)\}=\\ g^{ij}_a(u(x)) \delta^i(x- y)+ \Gamma^{ij}_{k;a}(u(x)) u^k_x\delta(x-y)+ \sum_{m\geq 1}\sum^{m+1}_{\ell= 0} \varepsilon^m A^{ij}_{m,\ell,a}(u, u_x,\dots, u^{(m+1-\ell}))\delta^{(\ell)}(x- y),\end{split} \] where \(A^{ij}_{m,\ell,\alpha}\) are homogeneous differential polynomials of degree \(m+1-\ell\), and the coefficients of these polynomials are smooth functions of \(u^1,\dots, u^n\).

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alber, M.S.; Camassa, R.; Fedorov, Yu.N.; Holm, D.D.; Marsden, J.E., The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDEs of shallow water and Dym type, Commun. math. phys., 221, 197-227, (2001) · Zbl 1001.37062
[2] Camassa, R.; Holm, D.D., An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 1661-1664, (1993) · Zbl 0972.35521
[3] Camassa, R.; Holm, D.D.; Hyman, J.M., A new integrable shallow water equation, Adv. appl. mech., 31, 1-33, (1994) · Zbl 0808.76011
[4] Casati, P.; Pedroni, M., Drinfeld – sokolov reduction on a simple Lie algebra from the Bihamiltonian point of view, Lett. math. phys., 25, 89-101, (1992) · Zbl 0767.58019
[5] G. Carlet, B. Dubrovin, Y. Zhang, The extended Toda hierarchy, nlin-SI/0306060, Moscow Math. J. 4 (2) (2004) 313.
[6] Dedecker, P.; Tulczyjev, W.M., Spectral sequences and the inverse problem of the calculus of variations, Lecture notes math., 836, 498-503, (1980)
[7] L. Degiovanni, F. Magri, V. Sciacca, On deformation of Poisson manifolds of hydrodynamic type. nlin.SI/010352. · Zbl 1108.53044
[8] Dickey, L.A., Soliton equations and Hamiltonian systems, (), 71 · Zbl 0753.35075
[9] B. Dubrovin, Geometry of 2D topological field theories, in: M. Francaviglia, S. Greco (Eds.), Integrable Systems and Quantum Groups, Montecatini, Terme, 1993. Springer Lecture Notes in Mathematics 1620 (1996) 120-348. · Zbl 0841.58065
[10] Dubrovin, B.; Novikov, S.P., The Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the bogolyubov – whitham averaging method, Sov. math. dokl., 270, 4, 665-669, (1983) · Zbl 0553.35011
[11] Dubrovin, B.; Novikov, S.P., On Poisson brackets of hydrodynamic type, Sov. math. dokl., 279, 2, 294-297, (1984) · Zbl 0591.58012
[12] Dubrovin, B.; Novikov, S.P., Hydrodynamics of weakly deformed soliton lattices. differential geometry and Hamiltonian theory, Russ. math. surv., 44, 35-124, (1989) · Zbl 0712.58032
[13] Dubrovin, B.; Zhang, Y., Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation, Commun. math. phys., 198, 311-361, (1998) · Zbl 0923.58060
[14] B. Dubrovin, Y. Zhang, Normal forms of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. math.DG/0108160.
[15] B. Dubrovin, Y. Zhang, Virasoro symmetries of the extended Toda hierarchy, math.DG/0308152, Commun. Math. Phys. 250 (2004) 161-193. · Zbl 1071.37054
[16] Eguchi, T.; Yamada, Y.; Yang, S.-K., On the genus expansion in the topological string theory, Rev. math. phys., 7, 279-309, (1995) · Zbl 0837.58043
[17] Ferapontov, E.V., Compatible Poisson brackets of hydrodynamic type, J. phys. A, 34, 11, 2377-2388, (2001) · Zbl 1010.37044
[18] Fokas, A.S., On a class of physically important integrable equations, Physica D, 87, 145-150, (1995) · Zbl 1194.35363
[19] Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the camassa – holm equation, Physica D, 95, 229-243, (1996) · Zbl 0900.35345
[20] Fuchssteiner, B.; Fokas, A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4, 47-66, (1981) · Zbl 1194.37114
[21] Gardner, C.S., Korteweg – de Vries equation and generalizations IV, J. math. phys., 12, 1548-1551, (1971) · Zbl 0283.35021
[22] Getzler, E., A Darboux theorem for Hamiltonian operators in the formal calculus of variations, Duke math. J., 111, 535-560, (2002) · Zbl 1100.32008
[23] Getzler, E., The Toda conjecture, (), 51-79 · Zbl 1047.37046
[24] Lichnerowicz, A., LES varietes de Poisson et leurs algèbres de Lie associeés, J. diff. geom., 12, 253-300, (1977) · Zbl 0405.53024
[25] Lorenzoni, P., Deformations of Bihamiltonian structures of hydrodynamic type, J. geom. phys., 44, 331-375, (2002) · Zbl 1010.37041
[26] Maltsev, A., The conservation of the Hamiltonian structures in whitham’s method of averaging, izvestiya, Mathematics, 63, 6, 1171-1201, (1999) · Zbl 0969.37031
[27] Magri, F., A simple construction of integrable systems, J. math. phys., 19, 1156-1162, (1978) · Zbl 0383.35065
[28] Mokhov, O.I., Compatible and almost compatible pseudo-Riemannian metrics, Funct. anal. appl., 35, 2, 100-110, (2001) · Zbl 1005.53016
[29] Zakharov, V.E.; Faddeev, L.D., Korteweg – de Vries equation is a completely integrable Hamiltonian system, Funkz. anal. priloz., 5, 18-27, (1971)
[30] Zhang, Y., On the \(C P^1\) topological sigma model and the Toda lattice hierarchy, J. geom. phys., 40, 215-232, (2002) · Zbl 1001.37066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.