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On non-local Hamiltonian operators of hydrodynamic type connected with Whitham’s equations. (English. Russian original) Zbl 0857.35108

Russ. Math. Surv. 50, No. 6, 1253-1255 (1995); translation from Usp. Mat. Nauk 50, No. 6, 165-166 (1995).
The author describes the non-local operators of hydrodynamic type of the form \[ A^{ij}=g^{ij}d- g^{is}\Gamma^j_{sk}u^k_x+ \sum^N_{\alpha=1} \varepsilon^\alpha (w^\alpha)^i_k u^k_x d^{-1}(w^\alpha)^j_i u^i_x, \] where \(g^{ij}\), \(\Gamma^j_{sk}\) \((w^\alpha)^i_j\) depend on \(\overline{u}= (u^1,\dots,u^n)\), \(\varepsilon^\alpha=\pm1\), and considers the family of diagonal metrics of the form \[ (g_n^{ii})^{-1}= {{(g_0^{ii})^{-1}}\over {(u^i)^n}},\quad (g_0^{ii})^{-1}=2 {\displaystyle{\text{ res}_{\lambda=u^i}}} \Biggl[{{dp}\over {d\lambda}}\Biggr]^2 d\lambda_i, \quad n=1,2,\dots,\;i=1,\dots,2q+1, \] where \(dp\) is the differential of the quasi-impulse arising in the theory of KdV equation. The Abelian differential of the second kind on the Riemannian surfaces is \[ w^2=\prod^{2g+1}_{i=1} (\lambda-u^i) \] with normalizing conditions \(\int^{u^{2i+1}}_{u^{2i}} dp=0\), \(i=1,\dots,g\). Based on the above discussion the author obtains two theorems about Hamilton operators.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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