# Instabilities Appearing in Effective Field theories: When and How?

Jean-Pierre Eckmann, Farbod Hassani, Hatem Zaag

Jean-Pierre Eckmann, Farbod Hassani, Hatem Zaag

**2/5/22**Published in : arXiv:2205.01055

Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation \partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2 +\beta \partial_x^2 u(x,t) in 1+1 dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when \alpha >0. We study the detailed nature of this divergence as a function of the parameters \alpha>0 and \beta\ge0. The divergence does not disappear even when \beta is very large contrary to what one might believe. But it will take longer to appear as \beta increases when \alpha is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to 3+1 dimensions.