I am working on solving a PDE using Lie Symmetries. The equation I have to solve is third order, and as such I need to find the third prolongation, $\eta^{xxx}$
I cannot find the explicit form of this prolongation, so I have had to determine $\eta^{xxx}$ myself.
I'd like to know if I have the correct expansion.
Here is what I have:
\begin{equation} \label{eq:n_x} \begin{split} \eta^{x} & = D_x \eta - u_x D_x \xi - u_t D_x \tau \\ & = \eta_x + \eta_u u_x - \xi_x u_x - \xi_u u^2_x - \tau_x u_t - \tau_u u_t u_x \\ & = \eta_x + (\eta_u - \xi_x) u_x - \xi_u u^2_x - \tau_x u_t - \tau_u u_t u_x \\ \end{split} \end{equation}
\begin{equation} \label{eq: n_xx} \begin{split} \eta^{xx} & = D_x (\eta^{x}) - u_{xx} D_x \xi - u_{xt} D_x \tau \\ & = \eta_{xx} + \eta_{xu} u_{x} + \eta_{xu} u_x + \eta_{uu} u^2_x + \eta_u u_{xx} \\ & - \xi_{xx} u_x - \xi_{xu }u^2_x - \xi_x u_{xx} - \xi_{xu} u^2_{x} - \xi_{uu} u^3_x - 2\xi_u u_x u_{xx} \\ & -\tau_{xx} u_t - \tau_{xu} u_x u_t - \tau_x u_{xt} - \tau_{xu}u_tu_x - \tau_{uu}u^2_xu_t - \tau_uu_tu_{xx} - \tau_uu_xu_{xt} \\ & - \xi_x u_{xx} - \xi_u u_x u_{xx} - \tau_x u_{xt} - \tau_u u_x u_{xt} \\ & = \eta_{xx} + (2 \eta_{xu} - \xi_{xx})u_x - \tau_{xx}u_t + (\eta_{uu} - 2\xi_{xu}) u^2_x - \xi_{uu}u^3_x - 2\tau_{xu}u_xu_t \\ & + (\eta_u - 2\xi_x)u_{xx} - 2\tau_x u_{xt} - 3\xi_u u_x u_{xx} - \tau_u u_t u_{xx} - 2\tau_u u_x u_{xt} \end{split} \end{equation}
\begin{equation} \label{eq:n_xxx} \begin{split} \eta^{xxx} & = D_x (\eta^{xx}) - u_{xxx} D_x \xi - u_{xxt} D_x \tau \\ & = \eta_{xxx} + (3 \eta_{xxu} - \xi_{xxx})u_x + (3\eta_{xuu} - 3\xi_{xxu})u^2_x + (\eta_{uuu} - 3\xi_{xuu})u^3_x - \xi_{uuu}u^4_x \\ & - \tau_{xxx} u_t - 3\tau_{xxu}u_xu_t - 3\tau_{xuu}u^2_xu_t - \tau_{uuu}u^3_xu_t - 3\tau_{xx}u_{xt} + (3\eta_{xu} - 3\xi_{xx})u_{xx} \\ & + (3\eta_{uu} - 9\xi_{xu})u_x u_{xx} - 6\xi_{uu}u^2_x u_{xx} - 6\tau_{xu}u_xu_{xt} - 3\tau_{uu}u^2_x u_{xt} -3\tau_{xu}u_t u_{xx} - 3\xi_u u^2_{xx} \\ & - 3\tau_u u_{xt}u_{xx} - 3\tau_{uu}u_x u_t u_{xx} + (\eta_u - 3\xi_x) u_{xxx} -4\xi_u u_x u_{xxx} - 3\tau_x u_{xxt} - 3\tau_u u_x u_{xxt} - \tau_u u_t u_{xxx} \end{split} \end{equation}
I know the first and second prolongations are correct.
Is the third prolongation also correct?
Thanks