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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)
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