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 ...

A chain of one-dimensional Schrödinger operators is called a “dressing chain” if they are connected by successive Darboux transformations. Particularly interesting are periodic dressing chains; they include finite-band operators and Painlevé equations as a special case. We investigate the Hamiltonian structure of these nonlinear lattices using V. Adler’s 2× 2 Lax pair. The Lax equation and the ...

This paper presents a new approach of constructing special ruled surfaces and aims to study their developability minimalist conditions. Our concept opens opportunities for application in engineering, surface modeling, architectural design. The principle our is introduce the original definitions Smarandache according Darboux frame curve lying on an arbitrary regular <math xmlns="http://www.w3.or...

The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see [8]. We prove that the universal character solves the Darboux chain. The N -periodic closing of the chain is equivalent to the Painlevé equation of type A N−1. Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters. Introduc...

1. An interesting class of real functions of a single real variable, the approximately continuous functions, was introduced by Denjoy [l] in his work on derivatives. The two principal facts discovered by Denjoy are that these functions are of Baire class 1 and have the Darboux property. Ridder [2] showed that the arguments of Denjoy apply to real functions of n variables. In this paper we discu...