论文标题
关于非线性schrödinger方程系统的标量型站立波解决方案
On scalar-type standing-wave solutions to systems of nonlinear Schrödinger equations
论文作者
论文摘要
在本文中，我们研究了一类非线性schrödinger方程系统的驻波解决方案。我们的目标是具有两个未知数的NLS系统的所有标准形式，它们具有共同的线性部分和立方规范的非线性，并产生具有强制动力学能量部分的哈密顿量。我们对基态的存在提供了必要的条件。此外，我们给出了基态形状的特征。事实证明，基态是标量型，即恒定向量的倍数和标量函数。我们进一步就相同形式的激发态的存在提供了足够的条件。还研究了基态的稳定性和不稳定。为此，我们对标量型站立溶液的研究介绍了一种抽象处理，该处理适用于具有均质能量 - 关键非线性的广泛类别的NLS系统。通过该论点，再现了一些先前的结果。
In this article, we study the standing-wave solutions to a class of systems of nonlinear Schrödinger equations. Our target is all the standard forms of the NLS systems, with two unknowns, that have a common linear part and cubic gauge-invariant nonlinearities and that yield a Hamiltonian with a coercive kinetic-energy part. We give a necessary and sufficient condition on the existence of the ground state. Further, we give a characterization of the shape of the ground state. It will turn out that the ground states are scalar-type, i.e., multiples of a constant vector and a scalar function. We further give a sufficient condition on the existence of excited states of the same form. The stability and the instability of the ground states are also studied. To this end, we introduce an abstract treatment on the study of scalar-type standing-wave solution that applies to a wide class of NLS systems with homogeneous energy-subcritical nonlinearity. By the argument, some previous results are reproduced.
