| Abstract: |
| The NLS system of the third-harmonic generation is analyzed. Our interest is in solitary wave solutions and their stability properties. The recent work of Oliveira and Pastor, discusses global well-posedness vs. finite time blow up, as well as other aspects of the dynamics. 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 stability. In dimension one, we show stability, whereas in dimensions two and three, they are generally spectrally unstable, except for a small region, where they do enjoy an extra pseudo-conformal symmetry.
Finally, we discuss the instability by blow-up. In the three dimensional case and for somewhat restricted set of parameters, we use virial identities methods to derive the strong instability. In the two dimensional case, the virial identities reduce matters, via conservation of mass and energy, directly to the initial data. Blow up is confirmed for all data, sufficiently close to the (unstable) soliton, with strictly larger mass. |
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