I'm trying to evaluate the evolution of two scalar fields but their equations of motion are coupled via a potential term $$ V(\phi, \psi) \supset \frac{1}{2}\lambda \phi^{2}\psi^{2}.$$
From the lagrangian, the equations of motion are:
$$ \ddot{\phi} - 3H\dot{\phi} + m^{2}\phi - \lambda \phi \psi^{2} = 0, $$
$$ \ddot{\psi} - 3H\dot{\psi} + m^{2}\psi - \lambda \psi \phi^{2} = 0, $$ where $H$ is the Hubble constant (although it depends on time, for my purpose it can be set to a constant) and the dots denote a differentiation in respect to time. I'm looking for a closed solution or a way to decouple those equations avoiding the obvious $\lambda = 0$ situation.
