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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 652076, 19 pages
http://dx.doi.org/10.1155/2012/652076
Research Article

A New Discrete Integrable System Derived from a Generalized Ablowitz-Ladik Hierarchy and Its Darboux Transformation

1Junior College, Zhejiang Wanli University, Ningbo 315100, China
2College of Mathematics, Honghe University, Mengzi 661100, China
3College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

Received 9 October 2011; Accepted 4 December 2011

Academic Editor: Zuo Nong Zhu

Copyright © 2012 Xianbin Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We find an interesting phenomenon that the discrete system appearing in a reference can be reduced to the old integrable system given by Merola, Ragnisco, and Tu in another reference. Differing from the works appearing in the above two references, a new discrete integrable system is obtained by the generalized Ablowitz-Ladik hierarchy; the Darboux transformation of this new discrete integrable system is established further. As applications of this Darboux transformation, different kinds of exact solutions of this new system are explicitly given. Investigatingthe properties of these exact solutions, we find that these exact solutions are not pure soliton solutions, but their dynamic characteristics are very interesting.

1. Introduction

It is well known that the nonlinear integrable lattice systems have been intensively investigated in many scientific fields such as physics, chemistry, and biology. The scientific researchers are not only interested in their rich mathematical structures, but also their applications in science, such as mathematical physics, numerical analysis, computer science, statistical physics, and quantum physics. Various integrable lattice systems have been studied extensively, such as the Toda lattice and the relativistic Toda lattice, the Ablowitz-Ladik lattice, the Volterra lattice, the Suris lattice, and some discrete classic equation such as discrete KdV equation, discrete Schrödinger equation, [118].

Recently, some new lattice hierarchies and their integrability, Darboux transformation, conversation law, exact solution, and dynamic characteristics have been holding more and more attention.In addition, the scheme for constructing nonlinear discrete integrable couplings [19] and new transformation methods which are used to directly solve discrete integrable systems [20] are also worth paying attention to. Among the multitudinous new integrable hierarchies, we will discuss the so-called new discrete lattice system which is given by [21, equation (2.12)]

which is derived by Qin. It is very interesting that this equation can be reduced to the old integrable lattice system under a series of local coordinate transformation, which is given by Merola et al. [22]. To see this,we give the whole transformation procedure as follows.

Taking the local coordinate transformation , (1.1) can be rewritten as

Taking another local coordinate transformation , (1.2) can be rewritten as

Taking the third coordinate transformation and , (1.3) can be rewritten as

To simplify notation, we still use and to take the place of and . Thus, (1.4) can be rewritten as

Equation (1.5)is equivalent to the following Equation:

where is a shift operator and . Equation (1.6) was given in [22] as the 44th equation. The results investigated by Qin in [21] are different from those given in [22]; however, these two systems are equivalent under the above coordinate transformations, noting that the coordinate scales of these two systems are different, and both work are interesting and very useful.

The rest of this paper is organized as follows. In Section 2, based on Qin’s work, we will derive an indeed new lattice system from the generalized Ablowitz-Ladik hierarchy and establish its Darboux transformation. In Section 3, under the Darboux transformation, using nonzero or zero constant as seeds, we will obtain different kinds of exact solutions of this new system and discuss their dynamic characteristics. It isworthy of mentioning that the exact solutions of this new lattice system which we will obtain have many particular phenomena; they are not pure soliton solutions and their waveforms are very interesting. In Section 4, brief conclusions are given.

2. New Discrete Lattice System and Its Darboux Transformation

As in [21], we introduce the following discrete spectral problem:

where

This is a generalization of the Ablowitz-Ladik spectral problem; its corresponding auxiliary problem is shown as follows:

First, we consider the stationary discrete zero curvature equation

where

Substituting (2.2) and (2.5) into (2.4), we obtain a series of recursive formulas as follows:

Second, we consider the discrete zero curvature equation with

where the positive power term and modification term are, respectively, given by

Substituting into the discrete zero curvature equation (2.7) and recurring to the stationary discrete zero curvature equation (2.3) yield

Thus, we obtain Obviously, when and , all equations in (2.11) satisfy compatibility condition between (2.1) and the following auxiliary spectral problem:

When and , letting and considering the integral constant as zero, by using (2.6) and (2.11), Qin derived (1.1). Moreover, when and ,,   letting and considering the integral constant as zero, by using (2.6) and (2.11) again,we successfully obtain an indeed new lattice system as follows:

Obviously, (2.13) is different from equation (2.12) appearing in [21], and their dynamic properties are very different, see the below discussion.

The Lax pairs of (2.13) are given by in (2.1) and in (2.12) when . The is given by

By using (2.6), when , substituting the values into the above matrix and comparing the coefficients of the different power of , we rewrite the matrix as

where

Next, we establish the Darboux transformation of this new discrete lattice system (2.13). Actually, the Darboux transformation is a special gauge transformation

of the solutions of the Lax pairs (2.1) and (2.12) when , where must satisfy Lax pairs (2.1) and (2.12) with some and , that is,

Meanwhile, it is required that and have the same forms as and , respectively. Thus, the old potentials in and are mapped into new potentials in and . We assume has the following form:

where are unknown functions which will be determined later.

Let and be two linear independent solutions of Lax pairs (2.1) and (2.12), and suppose that

Substituting (2.21) into (2.17), we obtain

We assume that and are two roots of . Therefore, when , the two columns of and are linear dependent, respectively. Thus, there exist two group of nonzero parameters and () which satisfy the following equations:

Solving the above equations, it follows that

where parameters and (note: ) are suitably chosen such that all the denominators in (2.24) and (2.25) are nonzero.

When , substituting (2.21) into (2.1), it follows,

Applying the shift operator to all expressions in (2.26) once, it follows

Solving (2.27) yields

By using (2.26)–(2.28), from (2.25), we get

where

According to the above analysis, we derive some useful functions in forthcoming computation process as follows:

We know that the matrix keeps the same form as in the procedure of mapping the old potentials in into new potentials in . We can suppose that

Substituting into the second formula in (2.18) and comparing the coefficients of the same power of , it follows that

From (2.36) and (2.38), it is easy to find that there exists a transformation which maps the old potentials into new ones as follows:

Respectively, substituting (2.41) into (2.34), (2.35), (2.37), (2.39), and (2.40), we obtain five conditional equations as follows:

Substituting (2.24)-(2.25), (2.29)-(2.30), and (2.31)-(2.32) into the above five conditional equations, taking a tedious direct computation or using the software Maple, we can verify all the above five equations are identities.

In fact, we have directly verified that the (2.41) is the Darboux transformation of the spectral problem (2.1) under the gauge transformation (2.17). Thus, the proof process such as those in [21] can be simplified. Next, we need to verify that the matrix defined by (2.19) has the same form as , that is,

where all the forms of , , , , are the same as the forms of , , , , . The matrices , , , , can be obtained as long as we use the and to take the place of and in the matrices , , , , , respectively.

To simplify computation, according to the coefficients of the different power of , we also rewrite the matrices as where

Substituting into the second formula in (2.19) yields

Letting the coefficients of the be zero, it follows that

So (2.48) holds. By dint of (2.41), we can easily verify that (2.49) and (2.50) hold. Under the transformation (2.41), substituting (2.24), (2.30), (2.31), and (2.32) into (2.51) and (2.52), taking a tedious direct computation or using the software Maple, we can verify that (2.51) and (2.52) hold. To simplify computation to verify (2.53) and (2.54), we need some other auxiliary conditions:

where

On the other hand,

Thus, we obtain

and their differentiation with respect to time

Applying (2.24), (2.30), (2.31), (2.32), (2.41), and (2.55)–(2.60), we can verify that (2.53) and (2.54) hold. Hence, the following theorem holds.

Theorem 2.1. The solutions of the discrete lattice system (2.13) are mapped into their new solutions under the Darboux transformation (2.41).

3. New Exact Solutions and Their Properties

In this section, we will construct exact solutions of discrete system (2.13) by using the Darboux transformation (2.41). Obviously, the are a couple of seed solutions of system (2.13). In addition, all the constants that satisfy conditional equation are seed solutions of system (2.13).Without loss of generality, we only consider two kinds of seed solutions, that is, and.

3.1. The Exact Solution Obtained by Nonzero Seed and Its Properties

When and , substituting the seed solutions into the Lax pairs (2.1) and (2.12), it follows that

where

 Equation (3.1) has two real linear independent basic solutions as follows:

where

By using (2.25), we obtain

where

Therefore, by using the Darboux transformation (2.41), (2.24), and (2.31), we obtain the new explicit solution of (2.13) as follows:where

which are reduced from (2.30); the parameters are defined by (3.6) and .

Equation (3.8) is a very complicated solution due to (3.6), and it is hard to analyze its dynamical behavior. Same as Xu, we did not know whether the26th solution given in [23] was a soliton solution. We also do not know whether the solution (3.8) is a soliton solution if we only estimate the type of solution from its expression. By the further detailed analysis, we know that the solution (3.8) is not a pure soliton solution, its properties and profiles vary accordingly while the parameters vary. In other words, under the different values of parameters, it contains different waveform. However, in [21], treating as continuous variable, Qin showed that solution (4.4) possesses properties of one-soliton solutions and two-soliton solutions when , and are taken as different parameters, respectively. In fact, the is a discrete variable, we cannot regard it as a continuous variable. The dynamic properties and profiles of solution would be distortion if we regard the discrete variable as a continuous variable. Therefore, regarding as a discrete variable, we will plot graphs of solution (3.8), some new properties and interesting phenomena will be produced. This shows that our analytic work on dynamic properties of exact solution obtained by the Darboux transformation is very different from the Qin’s work.

As an example, when are restricted in the region , we investigate the dynamical behavior of solution (3.8). The term in (3.6) can be rewritten as when , but it cannot be rewritten as this form when . It is for this reason that solution (3.8) has variety of properties. For example, when and are both held, solution (3.8) has no oscillation behavior. This can be seen in Figure 1.

fig1
Figure 1: The profiles of in (3.8) for .

Figure 1 shows the properties of double-kink soliton and single-kink soliton with singularity for solutions , respectively. Figures 1(a)1(c) show a procedure of leftwards moving for discrete double kink soliton. The waveform gradually changes from Figure 1(a) to Figure 1(c). Particulary, when , the waveform becomes a static singular kink soliton, which can be seen in Figure 1(b). Figures 1(d)1(f) also show a procedure of waveform gradually changes for discrete kink wave. The waveform is a discrete single kink soliton when the time , which can be seen in Figure 1(d). Particulary, when the time , the waveform also becomes a static singular kink soliton, which can be seen in Figure 1(e). When the time , the waveform becomes a moving singular kink wave, which can be seen in Figure 1(f).

When both and hold, the solution (3.8) is discrete breather solution; it has breather oscillation behavior shown in Figure 2.In fact, Figure 2 shows the properties of breather soliton with double-kink characteristic for solutions .

fig2
Figure 2: The profiles of in (3.8) for .

Figures 2(a)2(c) show a procedure of breather oscillation which moves from the right to the left as the time increases from −2 to 2. The waveform and amplitude gradually change from Figure 2(a) to Figure 2(c). Figures 2(d)2(f) also show a procedure of another breather oscillation which moves from the right to the left; the properties and profiles are very similar to those shown in Figures 2(a)2(c).

However, when (or ), solution (3.8) has no breather oscillation behavior anymore. In fact, Figure 3 shows some shapes of singular kink and anti-kink soliton for solutions .

fig3
Figure 3: The profiles of in (3.8) for .

Figures 3(a)3(c) show a procedure of singular antikink soliton which moves from the right to the left as the time increases from −2 to 3. The waveform and amplitude gradually change from the form in Figure 3(a) to that in Figure 3(c). Figures 3(d)3(f) also show a procedure of singular kink soliton which moves from the right to the left; the other properties and profiles are very similar to those shown in Figures 3(a)3(c) except for the static case at the time . The waveform in Figure 3(b) has singularity; however, this is not the case in Figure 3(e).

3.2. The Exact Solution Obtained by Zero Seed and Its Properties

When and , substituting the seed solutions into the Lax pairs (2.1) and (2.12), it follows that

Equation (3.10) has two real linear independent basic solutions as follows:

From (3.11), let ; by using (2.25) and (2.30),we obtain

Similarly, by using the Darboux transformation (2.41) and (2.24), (2.31), and (3.12), we obtain another explicit solution of (2.13) as follows:

In fact, solution (3.13) is not a pure soliton solution for arbitrary parametric values. By the further detailed analysis, we know that solution (3.13) becomes a soliton solution if one of the following parametric conditions holds:(i)(ii)(iii)(iv)

Under the above parametric conditions (i) or (ii), the solutions in (3.13) show discrete bright or dark soliton, which are shown in Figure 4. Under the above parametric conditions (iii) or (iv), the solutions in (3.13) show discrete kink or anti-kink soliton, which are shown in Figure 5.

fig4
Figure 4: The solutions of in (3.13) show a shape of discrete bright or dark soliton for fixed parametric values. Take (a) and (c) parameters as , and take (b) and (d) parameters as .
fig5
Figure 5: The solutions of in (3.13) show a shape of discrete kink or antikink soliton for fixed parametric values. Take (a) and (c) parameters as , and take (a) and (c) parameters as .

Under the other parametric conditions, solution (3.13) is not soliton, therefore, we omit those parts of graphs and discussions on dynamic properties.

4. Conclusions

In this paper, we show that the discrete system (1.1) which is given by Qin in [21] can be reduced to the old integrable lattice system which is given by Merola et al. [22]. Based on the Qin’s work, we present an indeed new discrete integrable system (2.13). Its Darboux transformation is established, and two complex exact solutions (3.8) and (3.13) are obtained by this transformation. By the further detailed analysis, we know that neither of these two exact solutions is pure soliton solutions. Their waveforms vary accordingly while the parameters vary. In other words, under different parametric conditions, these kinds of exact solutions show different dynamic behavior and profiles. In addition, by using different seeds, the obtained solutions are different, so are their properties and waveforms. The solutions obtained by nonzero constant seed mainly show the breather oscillation or singular behavior, but the solutions obtained by zero seed mainly show the profiles of discrete bright or dark soliton and kink or antikink soliton. These show that some exact solutions of discrete integrable systems obtained by the Darboux transformation are it is hard to know whether soliton solutions if we estimate only the types of solution from their expression. It is just because this reason that Xu did not know whether the solution obtained in [23] was a soliton solution. Perhaps, as in [24, 25], we call these kinds of solutions N-wave solutions or complexitons which is more accurate.Now, under what kind of parametric conditions, can the exact solutions of discrete integrable systems obtained by Darboux transformation show properties of soliton? This is a very interesting problem. We wish more and more researchers pay attention to this enthralling problem in the future works.

Acknowledgments

The authors thank three reviewers very much for their useful comments and helpful suggestions. This work was financially supported by the Natural Science Foundation of China (Grant no. 11161038). It was also supported by the Natural Science Foundations of Yunnan Province (Grant no. 2011FZ193) and Zhejiang Province (Grant no. Y2111160).

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