论文标题
轨道稳定性与散射在立方敏感的Schrodinger方程
Orbital stability vs. scattering in the cubic-quintic Schrodinger equation
论文作者
论文摘要
我们考虑了最多三个的空间尺寸中的立方Quintic非线性schr {Ö} dinger方程。因此，立方非线性是在焦点的同时集中精力的，从而确保了能量空间中凯奇问题的全球范围。本文的主要目的是研究孤立波的分散体和轨道稳定性之间的相互作用。 在空间维度一号中，已经知道所有孤子都轨道稳定。在第二个维度中，我们表明，如果初始数据属于保形空间，并且最多具有立方二维Schr {Ö} dinger方程的基质量的质量，则该溶液是渐近线性的。对于较大的质量，存在孤立的波解决方案，我们回顾了其稳定性的几个结果。最后，在第三维，依靠其他作者的先前结果，我们表明孤子可能是轨道稳定的，也可能不是轨道稳定的。
We consider the cubic-quintic nonlinear Schr{ö}dinger equation in space dimension up to three. 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 Schr{ö}dinger 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.
