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.

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