Darboux transformations and global explicit solutions for nonlocal Davey-Stewartson I equation

TL;DR

This paper develops Darboux transformations for the nonlocal Davey-Stewartson I equation, deriving explicit global solutions including solitons and line dark solitons, and analyzes their properties and singularity structures.

Contribution

It introduces a method to obtain explicit, global solutions for the nonlocal Davey-Stewartson I equation using Darboux transformations, extending the understanding of solution behaviors.

Findings

Derived explicit solutions including solitons and line dark solitons.

Proved solutions are global under specific parameter choices.

Analyzed the boundedness and peak structures of solutions.

Abstract

For the nonlocal Davey-Stewartson I equation, the Darboux transformation is considered and explicit expressions of the solutions are obtained. Like the nonlocal equations in 1+1 dimensions, many solutions may have singularities. However, by suitable choice of parameters in the solutions of the Lax pair, it is proved that the solutions obtained from seed solutions which are zero and an exponential function of $t$ respectively, by a Darboux transformation of degree $n$ are global solutions of the nonlocal Davey-Stewartson I equation. The derived solutions are soliton solutions when the seed solution is zero, in the sense that they are bounded and have $n$ peaks, and "line dark soliton" solutions when the seed solution is an exponential function of $t$, in the sense that they are bounded and their norms change fast along some straight lines.