Asymptotics for wild and large data in the nonconservative NLS

We study the nonconservative nonlinear Schrödinger equation (NLS) $u_t = idu_{xx} + iu^n$, $d>0$, and in contrast to most of the previous publications, we focus on the whole real line instead of periodic boundary conditions. Allowing for growth of the semigroup $e^{i\partial^2_x t}$ and based on a scaling argument, we show that solutions exist globally and admit decay to zero in $L^\infty$ when they start $L^\infty$-close to constants while the quantity $d^{\frac{1}{4}} ||\partial_xu(0)||_{L^2}$ is large. The second main result of this paper removes this $L^\infty$-smallness at the cost of localization assumptions.