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

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.


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.[1]).Jaulent-Miodek equation, Kaup-Newell equation, Levi equation and Heisenberg equation were found to be equivalent to AKNS equation [1]- [6].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.
[10]- [12] researched more general problems, and studied Hamilton structure and Darboux transformation and geometric algerba solutions.Ragnisco-Tu equation has spectral problem [7] , , 1 where ( ) , , E means a shift of space variable n, subscript t denotes partial derivative with respect to time t, λ is spectral parameter and , n n Q R 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 [13] [14].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.

The Derivation of Ragnisco-Tu Hierarchy
The derivation of Ragnisco-Tu hierarchy can be referred to [8] [9], 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 ( ) 2)-(2.5) can be written as where operators , 1, 2 , (2.9) 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 (2.12) and ( ) where I is an identity operator.In more general case, Ragnisco-Tu hierarchy is expressed by , ω λ ρ λ are the polynomials of λ with degree , 1 , and the boundary condition is as follows then general Ragnisco-Tu hierarchy adopts the from (2.15)

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 1 0 , , q q q r q r L r q q r r r we get matrix form ( ) ( ) be the polynomials of λ with degree k and ( ) the generalized Ragnisco-Tu hierarchy is Especially when ( ) 0 ω λ =  and ( ) , it is the iso-spectral hierarchy 1 .
If we take ( ) ( ) , then we get the non-iso-spectral hierarchy as follows 12) The first one (k = 0) is Proof.Expanding (3.7) we have ( ) .
Equating the coefficients of power of λ leads to ( ) Through mathematical induction we get the recursion relation From it the conclusion of Lemma 3.1 is got.

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.( ) Further, potentials in (1.2) and those in (1.3) have the relations , .
Notice that T is independent of λ , 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 , where 1 S T − = , we need to prove that equality N N =  is hold.A simple calculation shows us 11 12 where ( ) ( ) are the polynomial of λ with degree 1 k + , k , k , 1 k + respec- tively, which are the same as that of elements of N. In the meantime, the condition 0 and thus if permitting ( ) ( ) On the other hand, we can verify directly 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 12 N  as ( ) ( ) where 0,1, , 1 for iso-spectral and 1, 2, , 1 for non-iso-spectral.Because of in the case of iso-spectral and non-iso-spectral, the following recursion formula always holds ), we can deduce transformation relation by substituting the above relations into (4.7)( ) ( ) where Ω is the transfer operator of two hierarchies.Comparing the coefficients of 0 λ in 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 0 j = .That is 1 , which can be verified readily.When we concern about the non-iso-spectral case, (4.7) holds for 1 j = , the verification is not so easy.To get the equation 11 we first prove ( ) ( ) ( ) It is ease to know " 1 1 leftline rightline = ".As for " 2 line " we have n q E q E Enq q r q q r q r E q q r q r q r µ µ µ µ µ µ The Equation (4.11) is proved.
On the other hand, through comparing the coefficients of k λ in , That is ( ) ( ) Using the recursion relations of 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 ω in (4.3).The proof of part of ρ is similar and it is much simpler.Thus we finish the theorem.

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 [15]- [22].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.
Proof.Transformation T as DT must solve the following equation Comparing coefficients of , 0,1 j j λ = in entries of both side yields ( ) ( ) ( ) , , ( ) , , are two zeros of ( ) det H , then components of Hϕ are linear dependent, which means 0 Hϕ = .Thus j h can be determined according to formulas (5.3).Transformation (5.1) also change (2.1) to time evolution which matches to M .To justify this assertion, we first consider ( ) where

V h h h h h h A h h C h h B
and 2 3 , 0 h h → when n → ∞ we will find that ( )

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.

V 12 ,
polynomial of λ .As for asymptotic behavior of V 11 , obviously, as n → ∞ hold.In the mean time, h 1 , h 4 tend to 1 V 21 , V 22 , 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 0 R Q = = , we first obtain a solution to Riccati Equations (5Lemma 5.1, a solution to Ragnisco-Tu can be calculated out as follows

5 . 2 .
Darboux Transformation to Generalized Ragnisco-Tu EquationFrom gauge transformation, and DT of Ragnisco-Tu equation H ϕ ϕ = , we find relation between ψ and ψ : As a DT of generalized Ragnisco-Tu equation, P should satisfy V 22 and V 12 , V 21 are polynomials of λ ) into time evolution matching to M .Now we deal with V 11 as an example.First of all, referring to the fact that A n , D n and B n , C n are polynomials of λ with degree k + 1 and k, it is ease to know V 11 is polynomial of λ with degree k + 3. Secondly, accord-