Propagation of radius of analyticity for solutions to a fourth order
nonlinear Schr\”odinger equation
Abstract
We prove that the uniform radius of spatial analyticity
$\sigma(t)$ of solution at time $t$ to the
one-dimensional fourth order nonlinear Schr\”odinger
equation $$
i\partial_tu-\partial_x^4u
=|u |^2u $$ cannot decay faster than $1/
\sqrt{t}$ for large $t$, given initial data that is
analytic with fixed radius $\sigma_0$. The main
ingredients in the proof are a modified Gevrey space, a method of
approximate conservation law and a Strichartz estimate for free wave
associated with the equation.