Instabilities Appearing in Effective Field theories: When and How?
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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 ∂_t^2 u(x,t) = α (∂_x u(x,t))^2 +β∂_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 α >0. We study the detailed nature of this divergence as a function of the parameters α>0 and β≥0. The divergence does not disappear even when β is very large contrary to what one might believe. But it will take longer to appear as β increases when α 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.
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