Supersymmetric KdV equation: Darboux transformation and discrete systems

Abstract

For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the Bäcklund transformation found early by Liu and Xie [1]. The Darboux transformation and the related Bäcklund transformation are used to construct integrable super differential-difference and difference-difference systems. The continuum limits of these discrete systems and of their Lax pairs are also considered.

Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

“…Bäcklund transformations have been known to be an effective approach to construction of solutions for nonlinear systems, furthermore they may be applied to generate new integrable systems, both continuous and discrete [26,27,18]. It is remarked that the applications of Bäcklund transformations to integrable discretization of super or supersymmetric integrable systems were developed only recently [16,48,45,46,47,4,31].…”

On a supersymmetric nonlinear integrable equation in (2+1) dimensions

“…Since various techniques like Painlevé test [11], Darboux and Bäcklund transformations [6,8,19], Hirota bilinear method [7,13] and prolongation structure theory [15] have been extended to analysis supersymmetric integrable systems, a large number of (1+1)-dimensional integrable supersymmetric equations have been well studied, such as supersymmetric Korteweg-de Vries equation [5,12], supersymmetric Kadomtsev-Petviashvili hierarchy [10,18], supersymmetric nonlinear Schrödinger equation [14] and Heisenberg supermagnet model [4,9,21].…”

Super-QRT and 4D-mappings reduced from the lattice super-KdV equation

“…It is a coupled system of nonlinear discrete equations having two dependent variables with values in the commutative (bosonic) and anti-commutative (fermionic) sector of an infinite dimensional Grassmann algebra. The motivation comes from the recent construction of lattice super-KdV equation [17], where the Lax pair, consistency around the cube and super-multisoliton solution were constructed [1]. In this paper we consider the traveling wave reduction of the lattice super-KdV which gives an example of a super-QRT mapping as a fourth order mapping.…”