Review of well-posedness methods for the 1D nonlinear Schr\"odinger equation with an application to combined nonlinearities

Abstract

We consider the nonlinear Schr\"{o}dinger equation in one dimension with nonlinearities of type $|u|^\alpha u$ for any power $\alpha>0$ and review two different methods for obtaining solutions, namely, local well-posedness, with initial data either in $L^2$ or $H^1$, or in the weighted subspace of $H^1$. One approach is based on the Strichartz estimates, and thus, $H^1$ well-posedness typically holds for nonlinearities with power $\alpha \geq 1$. The other one is a direct application of weighted estimates commuting with derivatives and a certain infimum condition on the initial data, and thus, can treat nonlinearities for the whole range $0 < \alpha < \infty$; furthermore, it can handle a sum of different nonlinearities. We then conclude with an application of the second approach to the NLS with {\it finitely} many combined nonlinearities, important for physical applications (e.g., in laser optics), as it is more challenging, if at all possible, to obtain local well-posedness with the first method due to the lack of scaling invariance.