Is there any possible way to write this differential equation (1) in terms of planar dynamical system variables?
$\phi_{\xi \xi} = - \phi \frac{\phi_{\xi}^{2}(2 \phi^2 + 4 c) + (ck - \Omega^2) - 2 \phi^2 -4c}{(\phi^2 + 2c)^2},$ (1)
To simplify further, introduce $\phi^2 + 2c = \psi$, then above equation
$\psi_{\xi \xi} = \frac{(\frac{\psi^2}{2} - \phi^2 \psi)\psi_{\xi}^2 + 4 \psi -2(ck - \Omega)^2 \phi^4}{\phi^2 \psi^2},$ (2)
where $\phi=\phi(\xi),\;\psi=\psi(\xi)$, and all other parameters are contants. When introducing $\psi_{\xi} = \chi, \psi_{\xi \xi} = \chi_{\xi}$ in (2), it does not represent Hamiltonian system.
