Hamiltonian structures of the generalized cylindrical KdV equations (Q1124054)

\textit{H. H. Chen} and \textit{Guocheng Zhu} [J. Math. Phys. 27, 100-103 (1986; Zbl 0603.35081)] considered the following spectral problem \[ (1)\quad \phi_{xx}+u\phi -(x/(2t)+\lambda /4t)\phi =0;\quad \phi_ t=A\phi_ x+\beta \phi, \] where \(A=-(x/(3t)+2u+\lambda /t)\), \(B=1/(6t)+u_ x.\) Using \(\phi_{xxt}=\phi_{txx}\), the author in this paper gets the generalized cylindrical KdV equation \[ (2)\quad u_ t+u_{xxx}+6uu_ x+u/(2t)=0 \] and so proves (2) and its higher equations have Hamiltonian structures. In addition, the connection between the generalized C-KdV equation and nonisospectral problem is also pointed out.