On the cubic-quintic Schr\"odinger equation

TL;DR

This paper investigates the cubic-quintic Schr"odinger equation in Euclidean space, aiming to clarify unresolved issues and consolidate existing results, including bounds and existence of traveling waves.

Contribution

It provides an explicit constant for the $L^ Infty$ a priori bound and establishes partial existence results for finite energy traveling waves.

Findings

Explicit $L^ Infty$ a priori bound constant determined

Partial existence of finite energy traveling waves established

Consolidation of previous results on the cubic-quintic Schr"odinger equation

Abstract

This paper explores the cubic-quintic Schr\"odinger 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 $L^\infty$ a priori bound and establish a partial existence result for finite energy traveling waves in suitable approximate domains of $\mathbb R^d$.