A new Darboux transformation is presented for the Hirota-Satsuma coupled KdV system. It is shown that this Darboux transformation can be constructed by means of two methods: Painlevé analysis and reduction of a binary Darboux transformation. By iteration of the Darboux transformation, the Grammian type solutions are found for the coupled KdV system.

We consider the class of polynomial differential equation x&= , 2(,)(,)(,)nnmnmPxyPxyPxy++++2(,)(,)(,)nnmnmyQxyQxyQxy++&=++. For where and are homogeneous polynomials of degree i. Inside this class of polynomial differential equation we consider a subclass of Darboux integrable systems. Moreover, under additional conditions we proved such Darboux integrable systems can have at most 1 limit cycle.

The question of the correct formulation of one spatial problem of Darboux type for the wave equation has been investigated. The correct formulation of that problem in the Sobolev space has been proved for surfaces having a quite definite orientation on which are given the boundary value conditions of the problem of Darboux type. In the space of variables x1, x2, t we consider the wave equation ...

We use nonrelativistic supersymmetry, mainly Darboux transformations of the general (one-parameter) type, for the quantum oscillator thermodynamic actions. Interesting Darboux generalizations of the fundamental Planck and pure vacuum cases are discussed in some detail with relevant plots. It is shown that the one-parameter Darboux-transformed Thermodynamics refers to superpositions of boson and...

We study Darboux and Christooel transforms of isothermic surfaces in Euclidean space. These transformations play a signiicant role in relation to integrable system theory. Surfaces of constant mean curvature turn out to be special among all isothermic surfaces: their parallel constant mean curvature surfaces are Christooel and Darboux transforms at the same time. We prove | as a generalization ...

Sequences of Levy transformations for the Darboux system of conjugates nets in multidimensions are studied. We show that after a suitable number of Levy transformations, with at least a Levy transformation in each direction, we get closed formulae in terms of multiWroński determinants. These formulae are for the tangent vectors, Lamè coefficients, rotation coefficients and points of the surface...

We construct Darboux transformations for the super-symmetric KP hierarchies of Manin–Radul and Jacobian types. We also consider the binary Darboux transformation for the hierarchies. The iterations of both type of Darboux transformations are briefly discussed.

One of the important components of the theory of nonlinear S-integrable differential equations [1] is Darboux transformations, the story of which starts with the paper by Moutard [2] and therefore, contrary to the common tendency, we will call them Moutard-Darboux transformations. An example of an equation which admits Moutard-Darboux transformations is the second order linear differential equa...

Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides non auto-Bäcklund transformation between two n-th KdV equations with self-consistent sources with different degrees. The formula for the m-times repeated binar...