Global Analysis for Rough Solutions to the Davey-Stewartson System

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is in Hs with s > 2/5, then there exists a global solution in time, and the Hs norm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-time Lt,x4 estimate for nonlinear equation with the relatively general defocusing power nonlinearity

Nonlinear dynamics: challenges and perspectives

The study of nonlinear dynamics has been an active area of research since 1960s, after certain path-breaking discoveries, leading to the concepts of solitons, integrability, bifurcations, chaos and spatio-temporal patterns, to name a few. Several new techniques and methods have been developed to understand nonlinear systems at different levels. Along with these, a multitude of potential applications of nonlinear dynamics have also been enunciated. In spite of these developments, several challenges, some of them fundamental and others on the efficacy of these methods in developing cutting edge technologies, remain to be tackled. In this article, a brief personal perspective of these issues is presented

Traveling water waves — the ebb and flow of two centuries

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions