The technique of Darboux transformation is applied to nonlocal partner of two–dimensional periodic A n−1 Toda lattice. This system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with quantized spectral parameter. The generalization of the Darboux transformation technique on linear equations of such a kind is given. The solutio...

For a generalized super KdV equation, three Darboux transformations and the corresponding Bäcklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax representations. The reduction of one of the Bäcklund–Darboux transformations and the corresponding discrete system are considered for Kupershmidt’s super KdV equation....

In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the “invariance” of the objects of the “Darboux theory of integrability”. In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. ...

In this paper, the third kind of Darboux transformation of generalized Broer–Kaup equations is derived from the corresponding spectral problem. By virtue of this Darboux transformation, new 2N -soliton solutions with parameters of the generalized Broer–Kaup equations are obtained. Although 2N is an even number, it is graphically shown that in the cases of N= 1 and N = 2 the obtained 2N -soliton...

With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux transformations for the general self ad-joint schemes with five and seven neighbouring points. We also introduce a distinguished discretization of the two-dimension...

Darboux transformation is reconsidered for the supersymmetric KdV system. By iterating the Darboux transformation, a supersymmetric extension of the Crum transformation is obtained for the Manin-Radul SKdV equation, in doing so one gets Wronskian superdeterminant representations for the solutions. Particular examples provide us explicit supersymmetric extensions, super solitons, of the standard...

Darboux transformations in one variable form the basis for the factorization methods and have numerous applications to geometry, nonlinear equations and SUSY quantum mechanics. In spite of this wide range of applications the theory of Darboux transformations in two variables and its elegant relationship to analytic complex functions has not been recognized in the literature. To close this gap w...

For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases o...

The singular manifold method from the Painlevé analysis can be used to investigate many important integrable properties for the nonlinear partial differential equations. In this paper, the two-singular-manifold method is applied to the (2+1)-dimensional Gardner equation with two Painlevé expansion branches to determine the Hirota bilinear form, Bäcklund transformation, Lax pairs and Darboux tra...

The Toda lattice hierarchy with self-consistent sources and their Lax representation are derived. We construct a forward Darboux transformation (FDT) with arbitrary functions of time and a generalized forward Darboux transformation (GFDT) for Toda lattice with self-consistent sources (TLSCS), which can serve as a non-auto-Bäcklund transformation between TLSCS with different degrees of sources. ...