Title: On scalar-type standing-wave solutions to systems of nonlinear Schrödinger equations Show Chinese title

Title: 关于非线性薛定谔方程组的标量型驻波解

Abstract: In this article, we study the standing-wave solutions to a class of systems of nonlinear Schr\"odinger 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. Show Chinese abstract

Abstract: 在本文中，我们研究一类非线性薛定谔方程组的驻波解。 我们的目标是所有标准形式的NLS系统，这些系统有两个未知数，具有共同的线性部分和三次规范不变的非线性，并且产生一个具有强制动能部分的哈密顿量。 我们给出了基态存在的必要且充分条件。 此外，我们给出了基态形状的特征。 结果表明，基态是标量型的，即常向量和标量函数的倍数。 我们进一步给出了相同形式激发态存在的充分条件。 基态的稳定性和不稳定性也得到了研究。 为此，我们引入了一种关于标量型驻波解的抽象处理方法，适用于具有齐次能量亚临界非线性的广泛类别的NLS系统。 通过这种方法，一些先前的结果被重现。