The AB equations and the $\bar\partial$-dressing method in semi-characteristic coordinates

TL;DR

This paper extends the $ar ext{ extbackslash d}$-dressing method to analyze the AB equations in semi-characteristic coordinates, providing explicit soliton solutions and discussing their asymptotic behaviors.

Contribution

It generalizes the dressing method for AB equations and derives explicit soliton solutions using properties of the Cauchy matrix.

Findings

Explicit soliton solutions for AB equations are obtained.

Asymptotic behaviors of N-soliton solutions are analyzed.

The method provides a canonical form for AB equations.

Abstract

The dressing method based on the $2\times2$ matrix $\bar\partial$-problem is generalized to study the canonical form of AB equations. The soliton solutions for the AB equations are given by virtue of the properties of Cauchy matrix. Asymptotic behaviors of the $N$-soliton solution are discussed.