We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version – the Darboux transformations of Jacobi matrices – they are equivalent to Laurent polynomial modifications of the underlying measures. We address other questions which emphasize the similarities between Darboux transformations fo...

An isophote curve comprises a locus of the surface points whose normal vectors make a constant angle with a fixed vector. The main objective of this paper is to find the axis of an isophote curve via its Darboux frame and afterwards to give some characterizations about the isophote curve and its axis in Euclidean 3-space. Particularly, for isophote curves lying on a canal surface other characte...

We construct the generalized Darboux transformation with arbitrary functions in time t for the AKNS equation with self-consistent sources (AKNSESCS) which, in contrast with the Darboux transformation for the AKNS equation, provides a non-auto-Bäcklund transformation between two AKNSESCSs with different degrees of sources. The formula for N-times repeated generalized Darboux transformation is pr...

The theorem of unique solvability of a spatial problem of Darboux type in Sobolev space is proved for a second-order hyperbolic equation. In the space of variables x1, x2, t let us consider the second order hyperbolic equation Lu ≡ ƒu + aux1 + bux2 + cut + du = F, (1) where ƒ ≡ ∂ 2 ∂t2 − ∂ ∂x1 − ∂ 2 ∂x2 is a wave operator; the coefficients a, b, c, d and the right-hand side F of equation (1) ar...

We show that a recently developed modified Darboux transformation that uses foreign auxiliary equations, can be unified with the Darboux transformation for generalized Schrödinger equations. As a consequence of this unification, we obtain explicit Darboux transformations with foreign auxiliary equations of arbitrary order. © 2012 Elsevier Ltd. All rights reserved.

we consider the class of polynomial differential equation x&= , 2(,)(,)(,)nnmnmpxypxypxy++++2(,)(,)(,)nnmnmyqxyqxyqxy++&=++. for where and are homogeneous polynomials of degree i. inside this class of polynomial differential equation we consider a subclass of darboux integrable systems. moreover, under additional conditions we proved such darboux integrable systems can have at most 1 limit cycle.

We study the Darboux integrability of the celebrated Falkner– Skan equation f ′′′+ff ′′+λ(1−f ′2) = 0, where λ is a parameter. When λ = 0 this equation is known as Blasius equation. We show that both differential systems have no first integrals of Darboux type. Additionally we compute all the Darboux polynomials and all the exponential factors of these differential equations.

In this paper we discuss local and formal Darboux first integrals of analytic differential systems, using the theory of PoincaréDulac normal forms. We study the effect of local Darboux integrability on analytic normalization. Moreover we determine local restrictions on classical Darboux integrability of polynomial systems.

The paper contains a simple proof of the Finch-PatchRakesh inversion formula for the spherical mean Radon transform in odd dimensions. This transform arises in thermoacoustic tomography. Applications are given to the Cauchy problem for the Euler-Poisson-Darboux equation with initial data on the cylindrical surface. The argument relies on the idea of analytic continuation and known properties of...

In this Letter, we present the third kind of Darboux transformation of the classical Boussinesq system, and discuss its relationship with the two basic Darboux transformations. By applying the Darboux transformations, we obtain the solutions of multiple soliton interactions, including overtaking and head-on collisions.  2001 Published by Elsevier Science B.V.