Long time behavior of water waves: from asymptotic model to the full water waves

The local wellposedness of irrotational and incompressible water waves has been understood quite well since the breakthrough works of Wu, and the global existence for small and localized water waves was also established. Yet the understanding of the asymptotic behavior of the water waves is still limited, even for small initial data. In this talk, I will discuss the rigorous justification of the NLS from the full water waves, and its application to the study of modulational instability, including the rogue waves. We’ll see that, at least for wave packet like water waves, we can use the properties of the NLS to understand the long-time asymptotic behavior of the water waves.