In this talk, we will consider the NLS system of the third-harmonic generation. Our interest is in solitary wave solutions and their stability properties. The recent work of Oliveira and Pastor, discussed global well-posedness, finite time blow-up, as well as other aspects of the dynamics. These authors have also constructed solitary wave solutions, via the method of the Nehari manifold, in an appropriate range of parameters. Specifically, the waves exist only in spatial dimensions \(n=1,2,3\). They have also established 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 spectral stability.

Finally, we showed instability by a blow-up, for dimension 3, and for a more restrictive set of parameters, we use virial identities methods to derive the strong instability, in the spirit of Ohta's approach. This is joint work with Atanas Stefanov.