In this paper we present a trilinear form and a Darboux-type transformation to equation (ln v) xy = v − 1/v 2 considered by Tzitzéica in 1910. Soliton solutions are constructed by dressing the trivial solution.

Simple explicit formulae for finite time blow up solutions to the complex KdV equation are obtained via a Darboux transformation. Diffusions induced by perturbations are calculated. 2007 Elsevier Ltd. All rights reserved.

It has been recently discovered that exceptional families of SturmLiouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an examp...

The pulse propagation in the picosecond or femtosecond regime of birefringent optical fibers is governed by the coupled mixed derivative nonlinear Schrödinger (CMDNLS) equations. A new type of Lax pair associated with such coupled equations is derived from the Wadati–Konno–Ichikawa spectral problem. The Darboux transformation method is applied to this integrable model, and the N-times iteration...

Parasupersymmetry of the one-dimensional time-dependent Schrödinger equation is established. It is intimately connected with a chain of the time-dependent Darboux transformations. As an example a parasupersymmetric model of nonrelativistic free particle with threefold degenerate discrete spectrum of an integral of motion is constructed. 1. Supersymmetric quantum mechanics originally introduced ...

Let L be a quasi-definite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: xL, L + Cδ(x) and x L + Cδ(x) where δ(x) denotes the linear functional (δ(x))(xk) = δk,0, and δk,0 is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L....

In this paper we consider the Moutard transformation [9] which is a twodimensional version of the well-known Darboux transformation. We give an algebraic interpretation of the Moutard transformation as a conjugation in an appropriate ring and the corresponding version of the algebro-geometric formalism for two-dimensional Schrödinger operators. An application to some problems of the spectral th...