Symmetry Constraint of the Differential-difference KP Hierarchy and a Second Discretization of the ZS-AKNS System. (21st December 2017)

• Record Type: Journal Article
• Authors: Chen, Kui; Deng, Xiao; Zhang, Da-jun
• Is Part Of: Journal of nonlinear mathematical physics. Volume 24(2017)Supplement 1
• Page Start: 18
• Page End: 35
• Publication Date: 2017-12-21
• Subjects: differential-difference KP hierarchy -- squared-eigenfunction -- symmetry -- symmetry constraint -- ZSAKNS spectral problem -- semi-discrete AKNS hierarchies; 35Q51 -- 37K60; Nonlinear theories -- Periodicals; Mathematical physics -- Periodicals; 515.252
• DOI: 10.1080/14029251.2017.1418051
• Languages: English
• ISSNs: 1402-9251

Abstract

Abstract : In this paper we construct a squared-eigenfunction symmetry of the scalar differential-difference KP hierarchy. Through a constraint of the symmetry, Lax triad of the differential-difference KP hierarchy is reduced to a known discrete spectral problem and a semidiscrete AKNS hierarchy. The discrete spectral problem corresponds to a bidirectional discretization of the derivatives φ 1; x and φ 2; x in the ZS-AKNS spectral problem and therefore it is a discretization of the later. The discrete spectral problem is also known as a Darboux transformation of the ZS-AKNS spectral problem. Isospectral and nonisospectral flows derived from the spectral problem compose a Lie algebra. Infinitely many symmetries of the nonisospectral hierarchy are obtained. By considering infinite dimensional subalgebras of the algebra and continuum limit of recursion operator, three semi-discrete AKNS hierarchies are constructed.