Sharp threshold of global existence for the generalized Davey-Stewartson system in $R^2$

Abstract

This paper is concerned with the generalized Davey-Stewartsonsystem in $\mathbf R^2$ which appears as mathematical models for the evolution ofshallow-water waves having one predominant direction of travel. We obtaina sharp threshold of blowing up and global existence to the Cauchy problemof the system by constructing a type of cross-constrained variational problemand establishing so-called cross-invariant manifolds of the evolution flow. Furthermore, we answer the question: How small are the initial data, the globalsolutions to the Cauchy problem of the system exist.