We consider the NLS system of the third-harmonic generation, which was introduced in Sammut et al. (J Opt Soc Am B 15:1488–1496, 1998.). Our interest is in solitary wave solutions and their stability properties. The recent work of Oliveira and Pastor (Anal Math Phys 11, 2021), discussed global well-posedness vs. finite time blow up, as well as other aspects of the dynamics. These authors have also constructed solitary wave solutions, via the method of mountain pass/Nehari manifold, in an appropriate range of parameters. Specifically, the waves exist only in spatial dimensions \(n=1,2,3\). They have also establish some stability/instability results for these waves. In this work, we systematically build and study solitary waves for this important model. We construct the waves in the largest possible parameter space, and we provide a complete classification of their stability. In dimension one, we show stability, whereas in \(n=2,3\), they are generally spectrally unstable, except for a small region, where they do enjoy an extra pseudo-conformal symmetry. Finally, we discuss instability by blow-up. In the case \(n=3\), and for more restrictive set of parameters, we use virial identities methods to derive the strong instability, in the spirit of Ohta’s approach, Ohta (Funkc Ekvacioj 61(1):135–143, 2018). In \(n=2\), the virial identities reduce matters, via conservation of mass and energy, to the initial data. Our conclusions mirror closely the well-known results for the scalar cubic focussing NLS, while the proofs are much more involved.

我们考虑三次谐波生成的 NLS 系统，该系统在 Sammut 等人中介绍。 （J Opt Soc Am B 15：1488–1496，1998。）。我们感兴趣的是孤立波解决方案及其稳定性能。 Oliveira 和 Pastor 最近的工作（Anal Math Phys 11, 2021）讨论了全局适定性与有限时间爆炸，以及动力学的其他方面。这些作者还通过山口/Nehari 流形方法在适当的参数范围内构造了孤立波解。具体来说，波仅存在于空间维度\(n=1,2,3\)。他们还为这些波建立了一些稳定性/不稳定结果。在这项工作中，我们系统地构建和研究了这个重要模型的孤立波。我们在尽可能大的参数空间中构造波，并提供其稳定性的完整分类。在第一维中，我们表现出稳定性，而在 \(n=2,3\) 中，除了一小部分区域外，它们通常在光谱上不稳定，他们确实享受额外的伪共形对称性。最后，我们通过爆炸来讨论不稳定性。在 \(n=3\) 的情况下，对于更严格的参数集，我们使用维里恒等式方法来推导强不稳定性，在Ohta 方法的精神，Ohta（Funkc Ekvacioj 61(1):135–143, 2018）。在 \(n=2\) 中，维里恒等式通过质量和能量守恒将物质还原为初始数据。我们的结论与标量三次聚焦 NLS 的众所周知的结果密切相关，而证明则更为复杂。