# Does this system of coupled second order differential equations have a closed form solution?

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.

• Why the $\supset$? Jul 22 at 20:09
• Because the potential has this term included, but it is not the only term on the potential, as we have the mass term too, anymays is just to express that there is interaction between fields besides the mass term. Jul 22 at 20:11
• That is a weird and nonstandard usage of the symbol from set theory. So you'd say $x + \sin x \supset x$? Really? Jul 22 at 20:13
• We do it a lot in physics, sorry. Jul 22 at 20:14
• Here, a recent example eq. (1) from this particular paper, but if you search long enough in the HEP-PHENO section you will find a lot more: arxiv.org/abs/2206.01462 Jul 22 at 20:33