A Gauge Transformation between Ragnisco-Tu Hierarchy and a Related Lattice Hierarchy

Abstract

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.

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Liu, Y. , Hu, C. and Dai, J. (2015) A Gauge Transformation between Ragnisco-Tu Hierarchy and a Related Lattice Hierarchy. Journal of Applied Mathematics and Physics, 3, 1282-1294. doi: 10.4236/jamp.2015.310157.

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 (1.1)

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 (1.2)

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 (1.3)

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) (2.1)

discrete zero curvature equation results in following equalities directly (2.2) (2.3) (2.4) (2.5)

From these equations we draw out relations between related quantities (2.6) (2.7)

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

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

(2.8)

where operators are defined as

(2.9)

Giving boundary condition

(2.10)

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

(2.11)

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

If the boundary condition is given as

(2.12)

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

(2.13)

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

(2.14)

then general Ragnisco-Tu hierarchy adopts the from

(2.15)

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

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

(3.1)

Then from discrete zero curvature equation, we have

(3.2a)

(3.2b)

(3.2c)

(3.2d)

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

(3.3)

and

(3.4)

where are independent of n. Introducing two operators

(3.5)

(3.6)

we get matrix form

(3.7)

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

(3.8)

the generalized Ragnisco-Tu hierarchy is

(3.9)

Especially when and, it is the iso-spectral hierarchy

(3.10)

The first one (k = 0) is

(3.11)

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

(3.12)

The first one (k = 0) is

(3.13)

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:

(4.1)

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

(4.2)

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

(4.3)

where is transfer operator defined as

Proof. As gauge transformation, T should satisfy

(4.4)

Set

(4.5)

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

(4.6)

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

(4.7)

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,),

(In the case of iso-spectral,), we can deduce transformation relation by substituting the above relations into (4.7)

(4.8)

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

(4.9)

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

(4.10)

we first prove

(4.11)

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

(4.12)

That is

(4.13)

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

(5.1)

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:

(5.2)

where are determined by

(5.3)

where and is solution of Riccati equation

(5.4)

Proof. Transformation T as DT must solve the following equation

(5.5)

Comparing coefficients of in entries of both side yields

(5.6)

(5.7)

(5.8)

(5.9)

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)

(5.10)

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

(5.11)

5.2. Darboux Transformation to Generalized Ragnisco-Tu Equation

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

(5.12)

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

(5.13)

We can adopt simple notation to write it

(5.14)

where

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

(5.15)

where

From this expression, we can draw the following equalities

(5.16a)

(5.16b)

(5.16c)

(5.17a)

(5.17b)

(5.17c)

(5.18a)

(5.18b)

(5.19a)

(5.19b)

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

(5.20)

and

(5.21)

Thus we get

and form (5.17c), we obtain

Substituting it into (5.21), is figured out

(5.22)

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.

NOTES

*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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