Liu, Si-Qi; Zhang, Youjin Deformations of semisimple bihamiltonian structures of hydrodynamic type. (English) Zbl 1079.37058 J. Geom. Phys. 54, No. 4, 427-453 (2005). 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 \(u^1,\dots, u^n\). Reviewer: Messoud A. Efendiev (Berlin) Cited in 3 ReviewsCited in 44 Documents 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) Keywords:bi-Hamiltonian structure; infinitesimal deformation; Poisson cohomology PDFBibTeX XMLCite \textit{S.-Q. Liu} and \textit{Y. Zhang}, J. Geom. Phys. 54, No. 4, 427--453 (2005; Zbl 1079.37058) 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. 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