In the present investigation, breathers and rogue waves on double-periodic background are successfully constructed by Darboux transformation using a plane wave seed solution. Firstly, for reverse-space-time derivative nonlinear Schrödinger equation is constructed. Secondly, periodic solutions, breathers, derived n-fold transformation. Thirdly, higher-order semi-degenerate addition, dynamic beha...

We consider discrete Dirac systems as an approach to the study of corresponding block Toeplitz matrices, which in many ways completes famous via Szegő recurrences and matrix orthogonal polynomials. prove analog Christoffel–Darboux formula derive asymptotic relations for reproducing kernel (using Weyl–Titchmarsh functions systems). These are expressed also terms matrices. case rational (and GBDT...

In this paper we investigate rational minimal surfaces { a special class of umbilic-free minimal surfaces with nite total curvature and Enneper type ends. We deene an iteration for Gauss maps and show that it can be used to produce innnitely many families of rational functions that yield rational minimal surfaces{the Schwarzian derivative plays an important role in the proof. We also investigat...

We study the Gerdjikov-Ivanov (GI) equation and present a standard Darboux transformation for it. The solution is given in terms of quasideterminants. Further, parabolic, soliton breather solutions GI are as explicit examples.

Firstly, we establish the relation between loop group method and gauge transformation for 3×3 spectral problem. Some novel solutions of long wave-short wave model are obtained by Darboux method. Besides, give analysis classification solution in detail.

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, ...

It is shown that for every Darboux function F there is a non-constant continuous function f such that F + f is still Darboux. It is shown to be consistent—the model used is iterated Sacks forcing—that for every Darboux function F there is a nowhere constant continuous function f such that F + f is still Darboux. This answers questions raised in [5] where it is shown that in various models of se...