Short CommunicationConstruction of solutions for an integrable differential-difference equation by Darboux–Bäcklund transformation
Introduction
Beginning from the original work of Fermi, Past and Ulam in the 1960s [1], the integrable differential-difference equations, treated as models of many physical phenomena, have aroused widespread concern. Many nonlinear integrable differential-difference equations have been proposed, for instance the Ablowitz–Ladik lattice system [2], the Toda lattice system [3], the relativistic Toda lattice system [4], Merola-Ragnisco-Tu lattice system [5] and so on [6], [7], [8]. Their various properties have been studied from different points of view [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Furthermore, the Darboux–Bäcklund transformation (DBT) is a purely algebraic, powerful way to generate solutions for the soliton equations. It also can be applied to integrable differential-difference equation [13], [14], [15], [16], [17], [18], [19].
In Ref. [6] we present an integrable different-difference equation:This differential-difference system has good mathematical structure. In Ref. [6], we have studied its corresponding integrable differential-difference hierarchy, Hamiltonian structure, Louville integrability, conservation laws and integrable coupling system.
This paper is organized as follows. In Section 2, we first establish a one-fold Darboux–Bäcklund transformation of the Eq. (1) by the aid of a proper gauge transformation matrix. Then as result of iterated N-times of one-fold Darboux–Bäcklund transformation, we derive N-fold Darboux transformation-Bäcklund transformation of the Eq. (1). In Section 3, as application of the obtained Darboux–Bäcklund transformation, two exact solutions of the system (1) are deduced. Some conclusions are given in the final Section.
Section snippets
One-flod Darboux–Bäcklund transformation
For a lattice function the shift operator E, the inverse of E are defined byIn Ref. [6], we presented the following discrete matrix spectral problemand the temporal evolution law of eigenfunction ϕn obey the differential equationIn Eqs. (3) and (4), is the eigenfunction vector, is potential vector. The compatibility condition of the Eqs. (3)
Exact solutions
In what follows, we will use the resulting Darboux–Bäcklund transformation (14) to generate the solutions of the Eq. (1).
First,we chose the seed solution of the lattice system (1), namely, the simple special solution, . Substituting this solution into the corresponding Lax pair, we get
Solving above two equations, we obtain
Conclusions
In present work, we first deduced a one-fold Darboux–Bäcklund transformation of the integrable different-difference equation system (1.1) by means of a suitable gauge transformation of the corresponding Lax pair. After that we derived N-flod Darboux–Bäcklund transformation, which is equivalent to N times application of one-fold Darboux–Bäcklund transformation. As application of Darboux–Bäcklund transformation, two exact one-fold solutions are given.In addition, similar to the Ref. [20], the
Acknowledgments
This work is supported by the Youth Science Foundation Project of China (Grant No. 11805114) and the Nature Science Foundation of Shandong Province of China (Grant No. ZR2014AM001).
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