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...

Abstract We extend the Eruguin result exposed in the paper ”Construction of the whole set of ordinary differential equations with a given integral curve” published in 1952 and construct a differential system in R which admits a given set of the partial integrals, in particular we study the case when theses functions are polynomials. We construct a non-Darboux integrable planar polynomial system...

The problem of computing quantum mechanical propagators can be recast as a computation Wilson line operator for parallel transport by flat connection acting on vector bundle wavefunctions. In this picture the base manifold is an odd dimensional symplectic geometry, or quite generically contact that viewed "phase-spacetime", while fibers are Hilbert spaces. This approach enjoys "quantum Darboux ...

Abstract The article investigates systems of differential-difference equations hyperbolic type, integrable in sense Darboux. concept a complete set independent characteristic integrals underlying Darboux integrability is discussed. A close connection found between and Lie–Rinehart algebras the system. It proved that system if only its both directions are finite-dimensional.

The discrete non-commutative Darboux system of equations with self-consistent sources is constructed, utilizing both the vectorial fundamental (binary Darboux) transformation and method additional independent variables. Then symmetric reduction presented. In order to provide a simpler version resulting we introduce $\tau/\sigma$ form (commutative) system. Our give, in continuous limit, classical

A multidimensional version of the first Darboux problem is considered for a model second order degenerating hyperbolic equation. Using the technique of functional spaces with a negative norm, the correct formulation of this problem in the Sobolev weighted space is proved. In a space of variables x1, x2, t let us consider a second-order degenerating hyperbolic equation of the type Lu ≡ utt − |x2...

We construct so-called Darboux matrices and fundamental solutions in the important case of generalised Hamiltonian (or canonical) systems depending rationally on spectral parameter. A wide class explicit is obtained this way. Interesting results for dynamical several variables their follow. For these purposes we use our version Bäcklund-Darboux transformation square roots corresponding matrix e...

The goal of this paper is to characterize the family of averages of comparable (Darboux) quasi-continuous functions.

In this article, we have proved the Darboux's theorem. This theorem is important to prove the Riemann integrability. We can replace an upper bound and a lower bound of a function which is the definition of Riemann integration with convergence of sequence by Darboux's theorem. We adopt the following rules: x, y are real numbers, i, j, k are natural numbers, and p, q are finite sequences of eleme...