ncts

Nonlinear dispersive equations are PDE's describing various phenomena of waves, which are governed by their dispersion and nonlinear interactions. Typical equations are the nonlinear Schrodinger equation and the KdV equation. The main and challenging part of the analysis is to estimate competition between the spreading effect of dispersion and the amplifying effect of nonlinearity. In the last decade, there was a lot of progress for global analysis of large solutions, enabling us to describe and compare many different types of behavior in each equation, such as scattering, blow-up and solitons. Besides some results as well as open question, I would like to explain three major ideas and their combinations in the analysis: exploiting global dispersion to localize nonlinear interactions, distinguishing resonance to extract essential nonlinear effects, and imposing symmetry to enhance the global dispersion.