Darboux transformation for the Manin-Radul supersymmetric KdV equation

Abstract

In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to the Korteweg-de Vries equation. Soliton type solutions are constructed by dressing the vacuum and we present some relevant plots. * On leave of absence from Beijing Graduate School, CUMT,

Binary nonlinearization of the super AKNS system under an implicit symmetry constraint

“…Such as Darboux transformation [15,16], Hamiltonian structures [17,18,19], and so on. In very recent years, nonlinearization of the super AKNS system has been studied in Ref.…”

Low regularity a priori bounds for the modified Korteweg-de Vries equation

“…We will here prove another result in this direction, giving an a priori bound in H s for − 1 8 < s < 1 4 in terms of the H s norm of the initial data but establishing no continuity. Our method is analogous to that in Christ-CollianderTao [2] and Koch-Tataru [11] dealing with the nonlinear Schrödinger equation (NLS) on R. The related problem for the KdV equation was considered by Liu [12]. We note that our proof also applies for s = − The weak solution is constructed as a weak limit of strong solutions.…”

Solitons for a supersymmetric seventh-order KdV equation by superbilinear approach

“…So far, many of the tools used in standard theory have been extended to this framework, such as Hirota direct method [6], [7], [8] , Bäcklund transformations [2], [8] , prolongation theory, the Hamiltonian formalism [9], [10] , the Grassmannian description [11], [12], [13] , τ functions [14] , Darboux transformations, and Lie group direct method [15], [16] . A very interesting fact is that the SUSY KdV equation of Mathieu [17], [18], [19], [20], [21] does not have three-supersoliton solutions for an arbitrary choice of solitary waves, although it possesses a Lax pair [4] . Only for a special combination of parameters does the equation admit N -soliton solutions.…”