The 2D cubic Klein-Gordon equation is approximated by the cubic NLS up to O(ε^2) error over times of order ε^{-2/(N+1)} with a (1+t)^N prefactor.
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Error estimates in the non-relativistic limit for the two-dimensional cubic Klein-Gordon equation
The 2D cubic Klein-Gordon equation is approximated by the cubic NLS up to O(ε^2) error over times of order ε^{-2/(N+1)} with a (1+t)^N prefactor.