具有点缺陷的NLS的约束能量最小化和基态
Constrained energy minimization and ground states for NLS with point defects
摘要：
本文研究了一维非线性薛定谔方程的基态，其中缺陷位于一个不动点。非线性是聚焦的，并且由亚临界功率组成。基态的概念可以用几种（通常是不等价的）方式来定义。我们将基态定义为具有相同质量的函数中能量泛函的最小值。这是NLS主要应用领域中有物理意义的定义。在这方面，我们证明了一个抽象的定理，重温集中紧性方法，这是适合于处理非线性系统的不均匀性。然后，我们将其应用到三个模型，描述三种不同类型的缺陷：δ势，δ素数相互作用，和偶极子。在这三种情况下，我们明确地计算基态，我们显示他们的轨道稳定性。这个问题已经考虑了三角洲和三角洲素缺陷与不同的约束最小化问题，即定义基态作为Nehari流形上的行动的最小化。偶极缺陷的情况是全新的。
英文摘要：
We investigate the ground states of the one-dimensional nonlinear Schrödinger equation with a defect located at a fixed point. The nonlinearity is focusing and consists of a subcritical power. The notion of ground state can be defined in several (often non-equivalent) ways. We define a ground state as a minimizer of the energy functional among the functions endowed with the same mass. This is the physically meaningful definition in the main fields of application of NLS. In this context we prove an abstract theorem that revisits the concentration-compactness method and which is suitable to treat NLS with inhomogeneities. Then we apply it to three models, describing three different kinds of defect: delta potential, delta prime interaction, and dipole. In the three cases we explicitly compute ground states and we show their orbital stability. This problem had been already considered for the delta and for the delta prime defect with a different constrained minimization problem, i.e. defining ground states as the minimizers of the action on the Nehari manifold. The case of dipole defect is entirely new.
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