On the cubic-quintic Schrödinger equation

This paper explores the cubic-quintic Schrödinger equation in the entire Euclidean space. Our objectives are twofold: first, to advance the understanding of unresolved issues related to this equation, which are well known in the extensively studied Gross-Pitaevskii equation. Second, to consolidate existing results on the cubic-quintic equation, providing partial contributions. Specifically, we determine the explicit constant for the $L^{\mathrm{\infty }}$ a priori bound and establish a partial existence result for finite energy traveling waves in suitable approximate domains of ${\mathbb{R}}^d$.