A Darboux transformation is constructed for the modified Veselov-Novikov equation .

We present algebraic construction of Darboux matrices for 1+1dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux-Bäcklund transformation, based on different λ-dependencies of the Darboux matrix: polynomial, sum of partial fractions, or the transfer matrix form. We derive symme...

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.

We analyze a certain class of integral equations associated with Marchenko equations and Gel’fand-Levitan equations. Such integral equations arise through a Fourier transformation on various ordinary differential equations involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unper...

A version of the iterated Bäcklund-Darboux transformation, where Darboux matrix takes a form of the tranfer matrix function from the system theory, is introduced for the discrete canonical system and nonAbelian Toda lattice. Solutions are constructed and several spectral results are obtained.

Given any background (or seed) solution of the nonlinear Schrödinger equation, the Darboux transformation can be used to generate higher-order breathers with much greater peak intensities. In this work, we use the Darboux transformation to prove, in a unified manner and without knowing the analytical form of the background solution, that the peak height of a high-order breather is just a sum of...

It is proved that the Darboux transformation of the system of coherent states of a free particle leads to the states that may be treated as coherent states of soliton-like potentials.

Darboux transformation is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the once Darboux transformation yields the variable separable solutions which can be obtained from the truncated Painlevé analysis and the twice Darboux transformation leads to some new variable separable ...

Darboux transformation of both Barut-Girardello and Perelomov coherent states for the timedependent singular oscillator is studied. In both cases the measure that realizes the resolution of the identity operator in terms of coherent states is found and corresponding holomorphic representation is constructed. For the particular case of a free particle moving with a fixed value of the angular mom...

A dressing technique is used to improve zero range potential (ZRP) model. We consider a Darboux transformation starting with a ZRP, the result of the " dressing " gives a potential with non-zero range that depends on a seed solution parameters. Concepts of the partial waves and partial phases for non-spherical potential are used in order to perform Darboux transformation. The problem of scatter...