Jean Gaston Darboux

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Darboux, Jean Gaston

 

Born Aug. 13, 1842, in Nîmes; died Feb. 23. 1917, in Paris. French mathematician. Member of the Paris Academy of Sciences (1884) and its permanent secretary from 1900. Corresponding member of the St. Petersburg Academy of Sciences (1895).

Darboux’s principal works are devoted to problems of differential geometry (the theory of surfaces and the theory of curvilinear coordinates): Lectures on the General Theory of Surfaces (vols. 1 -4, 1887–96) and Lectures on Orthogonal Systems and Curvilinear Coordinates (1898). His research on geometry led him to investigate various problems of integrating differential equations. Important among his works in other branches of mathematics are the memoirs on the theory of integration and the theory of analytic functions and also his research on the problem of the expansion of functions in orthogonal functions, especially in Jacobi polynomials. In mechanics, Darboux engaged in fruitful work on various problems of kinematics, equilibrium, and small fluctuations of a system of points.

REFERENCES

Liapunov, A. M. “Gaston Darbu.” Iz. AN, 1917. vol. 11, pp. 351–52.
Hilbert, D. “Gaston Darboux (1842–1917).” Acta mathematica, 1920, vol. 42, pp. 269–73.
References in periodicals archive ?
Teodoru, The Darboux Problem for third order hyperbolic inclusions via continuous selections for continuous multifunctions, Fixed Point Theory, An International Journal on Fixed Point Theory, Computation and Applications, House of the Book of Science, Cluj-Napoca, Volume 11 (2010), No.1, 133-142.
MARCELLAN, Darboux transformations and perturbation of linear functionals, Linear Algebra Appl., 384 (2004), pp.
In order to get exact solutions directly, many powerful methods have been introduced such as inverse scattering method [1], bilinear transformation [2], Baklund and Darboux transformation [3-5], tanh-sech method [6, 7], extended tanh method [8], Exp-function method [9-11], the sine-cosine method [12-14], the Jacobi elliptic function method [15], F-expansion method [16, 17], auxiliary equation method [18,19], bifurcation method [20-22], homotopy perturbation method [23], and homogeneous balance method [24, 25].
It is worth discussing the properties of discrete integrable systems, such as Darboux transformations [12, 13], Hamiltonian structures [14-16], exact solutions [17], and the transformed rational function method [18].
Benchohra, "Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay," Nonlinear Analysis: Hybrid Systems, vol.
Later, it is shown that there is a relation between exceptional orthogonal polynomials and the Darboux transformation [42, 43].
However, to our knowledge, the soliton and breather solutions of (1) have not been generated through N-fold Darboux transformation and the interaction characters between two breathers have not been analysed.
In the last three decades, various powerful methods have been presented, such as extended tanh method [1], homogeneous balance method [2], Lie group method [3], Wronskian technique [4, 5], Darboux transformation method [6], Hirotas bilinear method [7-10], and algebro-geometrical approach [11].
By a famous theorem of Darboux [4], near each point in (M, [omega]) there e[x.sub.i]st coordinates ([x.sub.1], [y.sub.1], ..., [x.sub.n], [y.sub.n]) in which [omega] has the form [[SIGMA].sup.n.sub.i=1] d[x.sub.i] [conjunction] d[y.sub.i], so symplectic manifolds have no local invariants, except the dimension.
Fox DM, Darboux F, Carrega P (2007) Effects of fire-induced water repellency on soil aggregate stability, splash erosion, and saturated hydraulic conductivity for different size fractions.
Latushkin: Spectral analysis of Darboux transformations for the focusing NLS hierarchy, Journal D' Analyse Mathematique, 93 (2004), 139-197.
BANGEREZAKO, Discrete Darboux transformation for discrete polynomials of hypergeometric type, J.