Darboux Method Research Articles

Effect of initial pulse shape modulation on spontaneous soliton formation in the NSE model

An initial pulse with fairly steep fronts whose evolution is described by the nonlinear Schrodinger equation, splits into soliton-like pulses (spontaneous soliton formation). The number of solitons formed in this process can be estimated by the number of spectral points of the associated linear Zakharov-Shabat problem for the initial pulse. Exact solutions of the Zakharov-Shabat problem are constructed for some classes of initial piecewise-continuous pulses by using the Darboux method. This allows us to estimate the effect of the shape of the initial pulse on the number of formed solitions and their parameters.

Asymptotics for the 4F3 polynomials

Asymptotic expansions are given for the 4 F 3 and 4 φ 3 orthogonal polynomials which generalize the classical orthogonal polynomials. The expansions are applied to determine the complex sets of convergence of series of these polynomials. The proof of the main asymptotic expansion uses a convexity argument which is especially well suited to estimating certain hypergeometric series and their integral analogs. As an alternative approach to the asymptotics, a uniform version of Darboux's method is described.

Bäcklund transformations, soliton solutions and wave functions of Kaup–Newell and Wadati–Konno–Ichikawa systems

Using the Bargmann–Darboux method, the Bäcklund transformations, n-soliton solutions and corresponding wave functions of the Kaup–Newell and Wadati–Konno–Ichikawa systems are obtained. These results culminate in an algebraic recursive procedure for the determination of multisoliton solutions and their wave functions of the derivative and mixed derivative nonlinear Schrödinger equations iQt+Qxx∓iα(‖Q‖2Q)x ±β‖Q‖2Q=0, α>0, β≥0.

Isometric embedding of two-dimensional manifolds of negative curvature by the Darboux method

Isometric embedding of two-dimensional manifolds of negative curvature by the Darboux method

Darboux functions and perturbative theory for the ground-state energy of two-electron atoms

The power series of the lowest eigenvalue epsilon ( lambda ) of the helium isoelectronic sequence, lambda being the coupling constant for the interaction between the two electrons, is summed by the ansatz of a simple analytic function of class D (Darboux). Stationarity criteria are used in constructing such a function, and the basic information needed is only a few terms of the perturbative power series. The ground-state energy is evaluated with great accuracy without using any free parameter. The Darboux method is used to give an estimate of the asymptotic coefficients of the perturbative expansion. The contrast between different empirical procedures and the self-consistency of the present one is remarkable.

The method of Darboux

The method of Darboux

The extension of Darboux's method to systems in involution of partial differential equations of arbitrary order in two independent variables

The extension of Darboux's method to systems in involution of partial differential equations of arbitrary order in two independent variables

A uniform treatment of Darboux's method

DARBOUX showed in his famous memoir, [1], that if a function F( t ) was analytic at t = 0 , and if on its corresponding circle of convergence, I t l = R , 0 < R < oo, it possessed only a finite number of singular points, all of which were algebraic in nature, then the rate of growth of its Maclaurin coefficients, F r could be deduced as n ~ oo. If these singular points, Re I~ Re t~ ... , were free to vary, it was assumed that they were restricted in such a manner that their essential configuration did not change as the Oj varied. In practice, their configuration was preserved by restricting the 0j to closed sub-intervals of [0, 2n] whose boundaries were independent of n. In this paper we consider the special case

Note on the Function F(a, b; c — n; z)

The generating function [equation] may be used to find an estimate of F(a, b; b - n; z) for large positive values of n. When the point t = 1 - 1/z lies outside the circle t = 1 the singularity t = 1 may be used to find an estimate by the method of Darboux [1] and the result is F(a, b; b - n; z) ~ 1.

On Darboux's Method of Solution of Partial Differential Equations of the Second Order

Proceedings of the London Mathematical SocietyVolume s2_14, Issue 1 p. 71-88 Papers published in the proceedings of the London mathematical society On Darboux's Method of Solution of Partial Differential Equations of the Second Order J. R. Wilton M.A., D.Sc., J. R. Wilton M.A., D.Sc.Search for more papers by this author J. R. Wilton M.A., D.Sc., J. R. Wilton M.A., D.Sc.Search for more papers by this author First published: 1915 https://doi.org/10.1112/plms/s2_14.1.71AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volumes2_14, Issue11915Pages 71-88 RelatedInformation