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.
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 , bilinear transformation , Baklund and Darboux
transformation [3-5], tanh-sech method [6, 7], extended tanh method , Exp-function method [9-11], the sine-cosine method [12-14], the Jacobi elliptic function method , F-expansion method [16, 17], auxiliary equation method [18,19], bifurcation method [20-22], homotopy perturbation method , 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 , and the transformed rational function method .
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 , homogeneous balance method , Lie group method , Wronskian technique [4, 5], Darboux
transformation method , Hirotas bilinear method [7-10], and algebro-geometrical approach .
By a famous theorem of Darboux
, 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.