Integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system. II

TL;DR

This paper develops a new integrable semi-discrete version of the Davey-Stewartson system, differing from previous models, by constructing Lax pairs for elementary flows and combining them, also linking to a semi-discrete Yajima-Oikawa system.

Contribution

It introduces a novel integrable semi-discretization of the Davey-Stewartson system and connects it to a semi-discrete Yajima-Oikawa system through a linear transformation.

Findings

Constructed Lax pairs for elementary flows

Demonstrated commutativity of flows

Linked semi-discrete Davey-Stewartson and Yajima-Oikawa systems

Abstract

This is a continuation of our previous paper arXiv:1904.07924, which is devoted to the construction of integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system; in this series of papers, we refer to a discretization of one of the two spatial variables as a semi-discretization. In this paper, we construct an integrable semi-discrete Davey-Stewartson system, which is essentially different from the semi-discrete Davey-Stewartson system proposed in the previous paper arXiv:1904.07924. We first obtain integrable semi-discretizations of the two elementary flows that compose the Davey-Stewartson system by constructing their Lax-pair representations and show that these two elementary flows commute as in the continuous case. Then, we consider a linear combination of the two elementary flows to obtain a new integrable semi-discretization of the Davey-Stewartson system. Using a linear transformation of the continuous independent variables, one of the two elementary Davey-Stewartson flows can be identified with an integrable semi-discretization of the $(2+1)$-dimensional Yajima-Oikawa system proposed in https://link.aps.org/doi/10.1103/PhysRevE.91.062902 .