Relation between hyperbolic Nizhnik-Novikov-Veselov equation and stationary Davey-Stewartson II equation

TL;DR

This paper explores the connection between the hyperbolic Nizhnik-Novikov-Veselov equation and the stationary Davey-Stewartson II equation, using Lax systems and Darboux transformations to find explicit soliton solutions.

Contribution

It establishes a link between these integrable equations and develops a method to construct explicit multi-soliton solutions.

Findings

Derived global $n$-soliton solutions that decay at infinity.

Proved solutions approach zero exponentially at spatial infinity.

Showed solutions form $n^2$ lumps of peaks asymptotically.

Abstract

A Lax system in three variables is presented, two equations of which form the Lax pair of the stationary Davey-Stewartson II equation. With certain nonlinear constraints, the full integrability condition of this Lax system contains the hyperbolic Nizhnik-Novikov-Veselov equation and its standard Lax pair. The Darboux transformation for the Davey-Stewartson II equation is used to solve the hyperbolic Nizhnik-Novikov-Veselov equation. Using Darboux transformation, global $n$-soliton solutions are obtained. It is proved that each $n$-soliton solution approaches zero uniformly and exponentially at spatial infinity and is asymptotic to $n^2$ lumps of peaks at temporal infinity.