A Darboux transformation for two new differential-integral equations in the Manakov hierarchy is given by using the gauge transformation between the corresponding matrix spectral problems. As an application of the Darboux transformation, soliton solutions of the two equations are obtained. 1. INTRODUCTION In the past few years, there are extensive development in the area of nonlinear mathematic...

Darboux transformation is constructed for superfields of the super sine-Gordon equation and the superfields of the associated linear problem. The Darboux transformation is shown to be related to the super Bäcklund transformation and is further used to obtain N super soliton solutions. PACS: 02.30.Ik PACS: 12.60.Jv mohsin [email protected] On study leave from PRD (PINSTECH) Islamabad, Pakistan mhassa...

We obtain isomonodromic transformations for Heun’s equation by generalizing Darboux transformation, and we find pairs and triplets of Heun’s equation which have the same monodromy structure. By composing generalized Darboux transformations, we establish a new construction of the commuting operator which ensures finite-gap property. As an application, we prove conjectures in part III.

Algebraic Bargmann and Darboux transformations for equations of a more general form than the Schrödinger ones with an additional functional dependence h(r) in the righthand side of equations are constructed. The suggested generalized transformations turn into the Bargmann and Darboux transformations for both fixed and variable values of energy and an angular momentum.

Two families of solutions of a generalized non-Abelian Toda lattice are considered. These solutions are expressed in terms of quasideterminants, constructed by means of Darboux and binary Darboux transformations. As an example of the application of these solutions, we consider the 2-periodic reduction to a matrix sine-Gordon equation. In particular, we investigate the interaction properties of ...

On donne dans cet article une interprétation géométrique des « douze surfaces de Darboux », qui apparaissent en appliquant façon répétée transformation simple à déformation isométrique infinitésimale d’une surface l’espace euclidien dimension trois. Cette est version différentielle la trialité, concernant les immersions totalement isotropes quadrique projective réelle 6 définie par forme quadra...

Using a quaternionic calculus, the Christoffel, Darboux, Goursat, and spectral transformations for discrete isothermic nets are described, with their interrelations. The Darboux and spectral transformations are used to define discrete analogs for cmc-1 surfaces in hyperbolic space and to obtain a discrete version of Bryant's Weierstrass type representation.

A general scheme is proposed for introduction of lattice and qdifference variables to integrable hierarchies in frame of ∂̄-dressing method . Using this scheme, lattice and q-difference Darboux-ZakharovManakov systems of equations are derived. Darboux, Bäcklund and Combescure transformations and exact solutions for these systems are studied.

The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota–Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.