We consider the cubic-quintic nonlinear Schrödinger equation of up to three space dimensions. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is already known that all solitons are orbitally stable. In dimension two, we show that if the initial data belong to the conformal space, and have at most the mass of the ground state of the cubic two-dimensional Schrödinger equation, then the solution is asymptotically linear. For larger mass, solitary wave solutions exist, and we review several results on their stability. Finally, in dimension three, relying on previous results from other authors, we show that solitons may or may not be orbitally stable.

中文翻译：

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三次五次薛定谔方程中的轨道稳定性与散射

我们考虑最多三个空间维度的三次五次非线性薛定谔方程。三次非线性因此聚焦，而五次非线性散焦，确保了柯西问题在能量空间中的全局适定性。本文的主要目的是研究孤立波的色散和轨道（非）稳定性之间的相互作用。在空间维度一中，已知所有孤子都是轨道稳定的。在二维中，我们证明如果初始数据属于共形空间，并且最多具有三次二维薛定谔方程的基态质量，则解是渐近线性的。对于更大的质量，存在孤立波解，我们回顾了几个关于它们稳定性的结果。最后，在第三维度，依靠其他作者以前的结果，