Abstract We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular motion of planar curves.

A noncommutative version of the KP equation and two families of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux and binary Darboux transformations. Additionally, it is shown that these solutions may also be verified directly. This approach is reminiscent of the wronskian technique used for the Hirota bilinear form of the...

The iterations are studied of the Darboux transformation for the generalized Schrödinger operator. The applications to the Dym and Camassa-Holm equations are considered.

Using as an illustrative example the p=1 operator-ordered Wheeler-DeWitt equation for a closed, radiation-filled Friedmann-Robertson-Walker universe, we introduce and discuss the supersymmetric double Darboux method in quantum cosmology. A one-parameter family of " quantum " universes and the corresponding " wavefunctions of the universe " for this case are presented. Many interesting results h...

We construct the Darboux-Bäcklund transformation for the sigma model describing static configurations of the 2-dimensional classical continuum Heisenberg chain. The transformation is characterized by a non-trivial normalization matrix depending on the background solution. In order to obtain the transformation we use a new, more general, spectral problem. PACS Numbers: 02.30.Ik, 03.50.Kk, 05.45Yv.

Concepts of supersymmetry and parasupersymmetry known for the one-dimensional stationary Schrödinger equation are generalized to the time-dependent equation. Our approach is based on differential transformation operators for the non-stationary Schrödinger equation called Darboux transformation operators and on chains of such operators. As an illustration new exactly solvable time-dependent pote...

In this paper, we derive a Darboux transformation of the Hirota and the Maxwell-Bloch (H-MB) system which is governed by femtosecond pulse propagation through an erbium doped fiber and further generalize it to the matrix form of the n-fold Darboux transformation of this system. This n-fold Darboux transformation implies the determinant representation of nth solutions of (E([n]),p([n]),η([n])) g...