JAMPJournal of Applied Mathematics and Physics2327-4352Scientific Research Publishing10.4236/jamp.2015.310157JAMP-60687ArticlesPhysics&Mathematics A Gauge Transformation between Ragnisco-Tu Hierarchy and a Related Lattice Hierarchy uqingLiu1*ChaoHu1JuanDai2School of Physics &amp; Mathematics, Changzhou University, Changzhou, ChinaInformation and Technique College of Changzhou, Changzhou, China* E-mail:yqmail321@163.com(UL);201020150310128212946 September 2015accepted 25 October 28 October 2015© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

A new lattice hierarchy related to Ragnisco-Tu equation is proposed and its gauge equivalence to Ragnisco-Tu equation is proven. As an application of gauge transformation, we construct Darboux transformation (DT) of this new equation through DT of Ragnisco-Tu equation. An explicit exact solution is presented as an example.

Ragnisco-Tu Hierarchy Gauge Transformation Transfer Operator Darboux Transformation
1. Introduction

Nonlinear integrable equations usually have some marvellous properties such as Hamilton structure and infinitely many conservation laws. There are close connections between many of these equations. For instance, the cerebrated KdV equation, modified KdV equation, and nonlinear Schrödinger equation are reduction of AKNS system. Boussinesq equation and derivative nonlinear Schrödinger equation are linked to the constraint of KP equation (c.f.  ). Jaulent-Miodek equation, Kaup-Newell equation, Levi equation and Heisenberg equation were found to be equivalent to AKNS equation  - . The relation between two equations plays an important role, which makes one tackle with relatively unfamiliar equations through relatively familiar equations. However, in a general survey, there is comparatively less research on relatedness of lattice soliton equations than that of continuous soliton equations. One of the reasons behind this actuality is the lack of related example. In this paper we put forward a pair of nonlinear integrable lattice equations and investigate some relations such as gauge equivalence relation and Darboux transformation between them. Utilizing the relation which has been found, we will obtain an exact solution of equation.

Ragnisco-Tu equation  

is an integrable lattice soliton equation. Ref.  discussed its Hamilton structure, and proved that its continuous limit may result in AKNS system. Ref.  obtained its inverse scattering transformation and exact solution. Ref.  - researched more general problems, and studied Hamilton structure and Darboux transformation and geometric algerba solutions. Ragnisco-Tu equation has spectral problem 

where, E means a shift of space variable n, subscript t denotes partial derivative with respect to time t, is spectral parameter and are potential functions. The derivation of equation hierarchy will be given in Section 2. This spectral problem can be generalized to

This method for constructing new lattice equation was first used in modified Toda equation   . Surprisingly, equations obtained via these two spectral equations are equivalent, but this feature does not appear on Toda equation. On this basis, we further discuss Darboux transformation of them. With the help of gauge transformation and Darboux transformation of Ragnisco-Tu equation, we get a Darboux transformation of new equation, which is complex and difficult to construct directly.

This paper is organized as follows: in Section 2 and Section 3, we deduce the general hierarchies of Ragnisco-Tu and related generalized lattice hierarchy respectively. In Section 4, we derive a gauge transformation and transfer operator of two hierarchies. Section 5 will contribute to the Darboux transformation of two equations. Finally, in Section 6, a conclusion is presented.

2. The Derivation of Ragnisco-Tu Hierarchy

The derivation of Ragnisco-Tu hierarchy can be referred to   , but for completeness we still give a concise version.

Consider time evolution corresponding to (1.2)

discrete zero curvature equation results in following equalities directly

From these equations we draw out relations between related quantities

where defined as and are constants independent of variable n. Let

the relations (2.2)-(2.5) can be written as

where operators are defined as

Giving boundary condition

and taking we may deduce the iso-spectral hierarchy as follows:

The case of k = 0 just gives Ragnisco-Tu Equation (1.1).

If the boundary condition is given as

and we get non-iso-spectral Ragnisco-Tu hierarchy

where I is an identity operator. In more general case, Ragnisco-Tu hierarchy is expressed by

Lemma 2.1. If are the polynomials of with degree respectively, , and the boundary condition is as follows

then general Ragnisco-Tu hierarchy adopts the from

3. A New Lattice Hierarchy Related to the Ragnisco-Tu Hierarchy

With regard to generalized spectral problem (1.3), introduce the time evolution

Then from discrete zero curvature equation, we have

It is ease to know that there only have three independent equations, for instance, (3.2b), (3.2c), (3.2d). Now, from them we work out

and

where are independent of n. Introducing two operators

we get matrix form

Set

the general lattice hierarchy (called generalized Ragnisco-Tu hierarchy) is deduced in

Lemma 3.1. Let and be the polynomials of with degree k and respectively, take, then under boundary condition

the generalized Ragnisco-Tu hierarchy is

Especially when and, it is the iso-spectral hierarchy

The first one (k = 0) is

If we take, then we get the non-iso-spectral hierarchy as follows

The first one (k = 0) is

Proof. Expanding (3.7) we have

Equating the coefficients of power of leads to

Through mathematical induction we get the recursion relation

From it the conclusion of Lemma 3.1 is got.

4. A Gauge Transformation and Transfer Operator between the Ragnisco-Tu Hierarchy and Generalized Ragnisco-Tu Hierarchy

In this section we will give the conclusion about gauge transformation and transfer operator between the Ragnisco-Tu hierarchy and generalized Ragnisco-Tu hierarchy.

Theorem 4.1. There exists a gauge transformation changing Lax pair of generalized Ragnisco-Tu hierarchy (1.3), (3.1) into Lax pair of Ragnisco-Tu hierarchy:

Further, potentials in (1.2) and those in (1.3) have the relations

When, , hierarchy (2.15) and hierarchy (3.9) satisfy

where is transfer operator defined as

Proof. As gauge transformation, T should satisfy

Set

the entries of it must meet the following equations

Notice that T is independent of, its entries are determined easily.

Transformation matrix T also changes time evolution (3.1) into (2.1). To justify this assertion, for a newly defined or, where, we need to prove that equality is hold. A simple calculation shows us

where

It is evident that, , , are the polynomial of with degree, , , respectively, which are the same as that of elements of N. In the meantime, the condition is equiva-

lent to and thus if permitting, then is valid.

On the other hand, we can verify directly that. Combining all discussed above we conclude that That is, the gauge transformation (4.1) conveys time evolution of (3.1) into that of (2.1).

Now we deduce transfer operator of two hierarchies. A dull calculation simplifies the expression of as

Thus we have

where for iso-spectral and for non-iso-spectral.

Because of in the case of iso-spectral and non-iso-spectral, the following recursion formula always holds

(In the case of iso-spectral,),

where is the transfer operator of two hierarchies. Comparing the coefficients of in we have

According to the derivation expressions of iso-spectral and non-iso-spectral equation we arrive at the relation of two hierarchies immediately

When we focus our attention on the iso-spectral case, (4.7) holds for. That is

which can be verified readily. When we concern about the non-iso-spectral case, (4.7) holds for, the verification is not so easy. To get the equation

we first prove

Denote

It is ease to know “”. As for “” we have

and

Their difference is

The Equation (4.11) is proved.

On the other hand, through comparing the coefficients of in we have

That is

Using the recursion relations of and together with (4.11), (4.12) and the formula of transformation operator (4.8) we finally obtain the recursion relation of non-iso-spectral (4.10).

Finally, we consider relevancy of two hierarchy. The time part of (4.3) has given in (4.9). The following equation is deduced according to (4.13) and (4.8)

Equation (4.11) together with above expression yields

Through mathematical induction we can prove the part of in (4.3). The proof of part of is similar and it is much simpler. Thus we finish the theorem.

5. Darboux Transformation to Generalized Ragnisco-Tu Equation

Darboux transformation is a very useful tool to obtain exact solutions of nonlinear integral equation. It plays role in every type of equations such as lattice equation, discrete equation and high dimensional integral equation  - . However, the construction of DT of complex system may still encounter difficulty. Here, we will have the aid of gauge transformation to consider DT of generalized Ragnisco-Tu equation.

5.1. Darboux Transformation to Ragnisco-Tu Equation

Consider transformation

where are independent of. We can see that it is a DT of Ragnisco-Tu hierarchy:

Lemma 5.1. (see also  ) Suppose as, then aforementioned transformation (5.1) is DT of Ragnisco-Tu hierarchy expressed by Lemma 2.1 with,. If, and, are used to denote potentials of spectral problem (1.2) and that of spectral problem transformed through formula (5.1) respectively, then the relations between them are formulized as follows:

where are determined by

where and is solution of Riccati equation

Proof. Transformation T as DT must solve the following equation

Comparing coefficients of in entries of both side yields

Suppose are two zeros of, then components of are linear dependent, which means. Thus can be determined according to formulas (5.3).

Transformation (5.1) also change (2.1) to time evolution which matches to. To justify this assertion, we first consider

where means adjoint matrix of H. The expressions of are as follows

We will prove that, are zeros of, and V11, V22 and V12, V21 are polynomials of with degree k + 3 and k + 2 respectively. This assures entries of matrix have the same degrees as that of N. Assume asymptotic condition

and when we will find that has the same asymptotic behavior. Therefore, because of also satisfy the same different equation, we say, i.e., transformation (5.1) change (2.1) into time evolution matching to.

Now we deal with V11 as an example. First of all, referring to the fact that An, Dn and Bn, Cn are polynomials of with degree k + 1 and k, it is ease to know V11 is polynomial of with degree k + 3. Secondly, according to the definition of we work out

Substituting them and (5.4) into V11 gives rise to

It is not difficult to check that Noticing that, we may say that in

is k + 1 power polynomial of. As for asymptotic behavior of V11, obviously, make

as hold. In the mean time, h1, h4 tend to,. These results assure. When it comes

to V12, V21, V22, the proof is similar, we do not repeat it. Now we finish the proof that (5.1) is a Darboux transformation of Ragnisco-Tu hierarchy.

As an application we present a exact solution to Ragnisco-Tu Equation (1.1). Starting from seed solution, we first obtain a solution to Riccati Equations (5.4)

Then according to Lemma 5.1, a solution to Ragnisco-Tu can be calculated out as follows

5.2. Darboux Transformation to Generalized Ragnisco-Tu Equation

From gauge transformation and DT of Ragnisco-Tu equation, we find relation between and:

which forms DT of generalized Ragnisco-Tu equation. Matrix admits the following form

We can adopt simple notation to write it

where

As a DT of generalized Ragnisco-Tu equation, P should satisfy

where

From this expression, we can draw the following equalities

The acquisition of solution of them must be combined with relation exhibited in Darboux matrix (5.13). Here we do not consider general formula of solution but present a special solution related to.

Notice that and the definitions of (5.3), we can find some simple relations easily

When seed solution is substituted into (5.16a)-(5.19b) we find following relations

These equalities produce

and

Thus we get

and form (5.17c), we obtain

Substituting it into (5.21), is figured out

6. Conclusion

We propose a lattice equation hierarchy related to Rangnisco-Tu hierarchy (generalized RT equation) and prove that it is equivalent to Rangnisco-Tu hierarchy itself. The transfer operator of two hierarchies is obtained. As an application of gauge transformation, we obtain a Darboux transformation of generalized RT equation and acquire an exact solution of this equation.

Acknowledgements

The authors feel grateful to pertinent opinions of reviewer and careful work of editors.

Cite this paper

YuqingLiu,ChaoHu,JuanDai, (2015) A Gauge Transformation between Ragnisco-Tu Hierarchy and a Related Lattice Hierarchy. Journal of Applied Mathematics and Physics,03,1282-1294. doi: 10.4236/jamp.2015.310157

NOTESReferencesChen, D.Y. (2006) Introduction of Soliton. Science Press, Beijing. (In Chinese)Chen, D.Y. and Zeng, Y.B. (1985) The Trans-formation Operator of Nonlinear Evolution Equations I. Chinese Annals of Mathematics, 6B, 71-82.Chen, D.Y. and Zeng, Y.B. (1985) The Transformation Operator of Nonlinear Evolution Equations III. Acta Mathematica, 28, 161-173. (In Chinese)Chen, D.Y. and Li, Y.S. (1986) The Transformation Operator of Nonlinear Evolution Equations IV. Acta Mathematica, 29, 127-134. (In Chinese)Chen, D.Y. and Li, Y.S. (1986) The Transformation Operator of Nonlinear Evolution Equations V. Acta Mathematica Sinica, New Series, 2, 343-356. http://dx.doi.org/10.1007/BF02564935Chau, L.L., Shaw, J.C. and Yen, H.C. (1992) Solving the KP Hierarchy by Gauge Transformations. Communications in Theoretical Physics, 149, 263-278. http://dx.doi.org/10.1007/bf02097625Zhang, H.W., Tu, G.Z., et al. (1991) Symmetries, Conserved Quantities, and Hier-archies for Some Lattice Systems with Soliton Structure. Journal of Mathematical Physics, 32, 1908-1918. http://dx.doi.org/10.1063/1.529205Zeng, Y.B. (1998) Restried Flows of a Hierarchy of Integrable Discrete Systems. Acta Mathematicae Applicatae Sinica, 14, 176-184. http://dx.doi.org/10.1007/BF02677424Liu, Y.Q. and Chen, D.Y. (2011) The Exact Solutions to a Ragnisco-Tu Hierarchy with Self-Consistent Sources. Nonlinear Analysis, 74, 5223-5237. http://dx.doi.org/10.1016/j.na.2011.04.021Xu, X.X. (2010) An Integrable Coupling Family of Merola-Ragnisco-Tu Lattice Systems, Its Hamiltonian Structure and Related Nonisospectral Integrable Lattice Family. Physics Letters A, 374, 401-410. http://dx.doi.org/10.1016/j.physleta.2009.11.012Zeng, X. and Geng, X.G. (2014) Algebro-Geometric Solutions of the Discrete Ragnisco-Tu Hierarchy. Reports on Mathematical Physics, 73, 17-48. http://dx.doi.org/10.1016/S0034-4877(14)60030-6Xu, X.X. (2015) Solving an Integrable Coupling System of Mer-ola-Ragnisco-Tu Lattice Equation by Darboux Transformation of Lax Pair. Communications in Nonlinear Science and Numerical Simulation, 23, 192-201. http://dx.doi.org/10.1016/j.cnsns.2014.11.002Xu, X.X., Yang, H.X. and Sun, Y.P. (2006) Dar-boux Transformation of the Modified Toda Lattice Equation. Modern Physics Letters B, 20, 641-648. http://dx.doi.org/10.1142/S0217984906011025Hon, Y.C. and Fan, E.G. (2009) Quasi-Periodic Solutions for Modified Toda Lattice Equation. Chaos, Solitons and Fractals, 40, 1297-1308. http://dx.doi.org/10.1016/j.chaos.2007.09.008Spiridonov, V. and Zhedanov, A. (1995) Discrete Darboux Transformations, the Discrete-Time Toda Lattice, and the Askey-Wilson Polynomials. Methods and Applications of Analysis, 2, 369-398.Fan, E.G. (2001) Explicit N-Fold Darboux Transformations and Soliton Solutions for Nonlinear Derivative Schr&#246;dinger Equations. Com-munications in Theoretical Physics, 35, 651-656. http://dx.doi.org/10.1088/0253-6102/35/6/651Zeng, Y.B., Shao, Y.J. and Ma, W.X. (2002) Integral-Type Darboux Transformations for the mKdV Hierarchy with Self-Consistent Sources. Communications in Theoretical Physics, 38, 641-648. http://dx.doi.org/10.1088/0253-6102/38/6/641He, J.S., Zhang, L., Chen, Y. and Li, Y.S. (2006) Determinant Representation of Darboux Transformation for the AKNS System. Science in China Series A: Mathematics, 49, 1867-1878. http://dx.doi.org/10.1007/s11425-006-2025-1Zhou, Z.X. (2007) Darboux Transformations for Some Two Dimensional Affine Toda Equations. The International Congress of Chinese Mathematicians (ICCM), III, 405-416.Wang, L., Gao, Y.T., Gai, X.L., Yu, X. and Sun, Z.Y. (2010) Vadermonde-Type Odd-Soliton Solutions for the Whitham-Broer-Kaup Model in the Shallow Water Small-Amplitude Regime. Journal of Nonlinear Mathematical Physics, 17, 197-211. http://dx.doi.org/10.1142/S1402925110000714Geng, X.G. and He, G.L. (2010) Darboux Transformation and Explicit Solutions for the Satsuma-Hirota Coupled Equation. Applied Mathematics and Computation, 216, 2628-2634. http://dx.doi.org/10.1016/j.amc.2010.03.107Xu, G.H. (2014) N-Fold Darboux Transformation of the Jaulent-Miodek Equation. Applied Mathematics, 5, 2657-2663. http://dx.doi.org/10.4236/am.2014.517254