A noncommutative version of the modified KP equation and a family of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux transformations and the solutions are verified directly. Furthermore, we present a noncommutative Miura transformation mapping solutions of the noncommutative mKP equation to solutions of the noncommutativ...

Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied. Bivariate linear functionals encompass large families of orthogonalities like Sobolev and discrete Sobolev types. Two possible Christoffel transformations of these linear functionals are discussed. Either the linear functionals are multiplied ...

We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analogue of the Koenigs net. We construct the Darboux-type transformation of the Koenigs lattice and we show permutability of superpositions of such transformations, thus proving integrability of the Koenigs lattice. We also investi...

In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the iterative rule and it can be expressed by the determinant form. In particular, I calculate first-order and second-order rational solutions and obtain their figures according to diffe...

A GBDT version of the Bäcklund–Darboux transformation is constructed for a non-isospectral canonical system, which plays essential role in the theory of random matrix models. The corresponding Riemann–Hilbert problem is treated and some explicit formulas are obtained. A related inverse problem is formulated and solved.

The Darboux transformations of Krawtchouk polynomials are investigated and all possible exceptional obtainable from a single-step transformation considered. properties these including the Diophantine ones recurrence relations obtained.

We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its ZakharovShabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the n-th Gel’fand-Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above mentioned ...

New types of irreducible second order Darboux transformations for the one dimensional Schrödinger equation are described. The main feature of such transformations is that the transformation functions have the eigenvalues grater then the ground state energy of the initial (or reference) Hamiltonian. When such a transformation is presented as a chain of two first order transformations, an interme...