A deformed reduced semi-discrete Kaup–Newell equation, the related integrable family and Darboux transformation
Introduction
Beginning from the original work of Fermi, Past and Ulam in the 1960s [1], the integrable differential–difference equations (or the lattice soliton equations) have received considerable attention. Many integrable differential–difference equations have been presented [2], [3], [4], [5], [6]. Much investigation on the integrable differential–difference equations has been obtained [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], such as the inverse scattering transformation [2], the symmetries and master symmetries [7], [8], [9], Hamiltonian structures [10], [11], [12], [13], [14], [15], [16], integrable coupling systems [13], [14], [15], [16], nonlinearization of the Lax pairs [17], [18], constructing complexiton solutions by the Casorati determinant [19], the Darboux transformations [20], [21], [22], and so on. For a function ,the shift operator E, the inverse of E and the difference operator D are defined byIn Ref. [23], a reduced semi-discrete Kaup–Newell equationis introduced, its r-Matrix and conserved quantities are presented. In Eq. (2), if we use instead of , and apply in the first equation, Eq. (2) becomesTherefore, Eq. (3) is a deformed reduced semi-discrete Kaup–Newell equation. In this letter we would like to research Darboux transformation of Lax pair of the deformed reduced semi-discrete Kaup–Newell equation (3). As is well known,Darboux transformation of Lax pair is an important method to find exact solutions of the integrable differential–difference equations. Unlike some complicated analytical methods, Darboux transformation is a powerful pure algebraic method. Furthermore, a remarkable feature of many integrable differential–difference equations is that they are members of integrable families, every family makes up of an infinite sequence of integrable differential–difference equations. The members in the same family share identical spatial part of Lax pair and can be solved by the same procedure via inverse scattering method. Therefore, it is important to find out which family the deformed reduced semi-discrete Kaup–Newell equation (3) belongs to. In what follows, we not only derive the Lax pair of Eq. (3), but also deduce the corresponding integrable family of Eq. (3).
This paper is organized as follows. In Section 2, we introduce a matrix spectral problem is eigenfunction vector, is the spectral parameter and is the potential vector, and depend on integer and real . Make use of the discrete zero curvature representation, we are going to derive a family of integrable differential–difference equations. In obtained family, the typical member is the deformed reduced semi-discrete Kaup–Newell equation (3). In Section 3, a Darboux transformation is constructed by means of the gauge transformation of Lax pairs for the deformed reduced semi-discrete Kaup–Newell equation (3). In Section 4, as applications of Darboux transformation, three exact solutions of Eq. (3) are given. Finally, in Section 5, there will be some conclusions and remarks.
Section snippets
The family of integrable differential–difference equations
In this section, we shall derive a family of integrable differential–difference equations associated with eigenvalue problem (4). To this end, we first solve the following stationary discrete zero curvature equationUpon settingWe find that Eq. (5)becomesSubstituting expansions
Darboux transformation
In what follows, we proceed to establish a Darboux transformation of Lax pair of the deformed reduced semi-discrete Kaup–Newell equation (3). It is well known that a gauge transformation of a matrix spectral problem is called a Darboux transformation if it transforms the matrix spectral problem into another spectral problem of the same type. We introduce the gauge transformationwhich can transform two spectral problems (4), (11) intowith
Exact solutions
In what follows, we will apply obtained Darboux transformation equation (27) to give exact solutions of Eq. (3).
First,it is easy to find that make up a trivial solution of Eq. (3).Substituting this solution into the corresponding Lax pair, we getSolving the above two equations, we get two real linear independent solutionsWe have
Conclusions and remarks
In this paper, we have deduced a deformed reduced semi-discrete Kaup–Newell equation and related integrable family through the discrete zero curvature equation. With the help of a gauge transformation of the Lax pair, a Darboux transformation is established for the deformed reduced semi-discrete Kaup–Newell equation, three exact solutions are given.Furthermore, starting from the above exact solutions, we apply the Darboux transformation (27) once again, then another new solutions is obtained.
Acknowledgments
The author is very grateful to the referees for the helpful suggestions.
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