While studying physics, I ended up having to find solutions for the following partial differential equation:
\begin{equation} \left[ \frac{1}{2}\left( \frac{\partial ^{2}}{\partial \alpha ^{2}}-\frac{% \partial ^{2}}{\partial \phi ^{2}}\right) +e^{6\alpha-\lambda k \phi }\right] \Psi (\alpha ,\phi )=0 \end{equation}
Where \begin{equation} \psi=R(\alpha, \phi)e^{iS(\alpha ,\phi)} \end{equation}
is a complex function and Vo and lambda are positive constants. The challenge is to find a analytical solution such that \begin{equation} R(0, \phi)=0 \end{equation}
By simply proposing \begin{equation} R(\alpha, \phi)=\alpha f(\phi) \end{equation}
I get something which seems untractable. A hint may be that, upon introducing u,v such that
\begin{equation} u=\frac{\sqrt[2]{2V_{0}}}{3}\frac{e^{3\alpha -(\lambda \sqrt[2]{6}/2)\phi }}{% 1-(\lambda /\sqrt[2]{6})^{2}}(\cosh (A)+\frac{\lambda }{\sqrt[2]{6}}senh(A)) \end{equation} and \begin{equation} v=\frac{\sqrt[2]{2V_{0}}}{3}\frac{e^{3\alpha -(\lambda \sqrt[2]{6}/2)\phi }}{% 1-(\lambda /\sqrt[2]{6})^{2}}(senh(A)+\frac{\lambda }{\sqrt[2]{6}}\cosh (A)) \end{equation}
where \begin{equation} A=3\phi -\frac{\lambda \sqrt[2]{6}}{2}\alpha \end{equation} the equation becomes just a klein-gordon equation \begin{equation} \left( \frac{\partial ^{2}}{\partial u^{2}}-\frac{\partial ^{2}}{\partial v^{2}}\right) \Psi (u,v)+\Psi (u,v)=0 \end{equation}
I´ve tried to solve it by imposing an ansatz for R in these new variables but it gets terribly ugly. Could there be any hope?
