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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.

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. [

Ragnisco-Tu equation [

is an integrable lattice soliton equation. Ref. [

where

This method for constructing new lattice equation was first used in modified Toda equation [

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 can be referred to [

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

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

where operators

Giving boundary condition

and taking

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

If the boundary condition is given as

and

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

Lemma 2.1. If

then general Ragnisco-Tu hierarchy adopts the from

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

we get matrix form

Set

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

Lemma 3.1. Let

the generalized Ragnisco-Tu hierarchy is

Especially when

The first one (k = 0) is

If we take

The first one (k = 0) is

Proof. Expanding (3.7) we have

Equating the coefficients of power of

Through mathematical induction we get the recursion relation

From it the conclusion of Lemma 3.1 is got.

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

where

Proof. As gauge transformation, T should satisfy

Set

the entries of it must meet the following equations

Notice that T is independent of

Transformation matrix T also changes time evolution (3.1) into (2.1). To justify this assertion, for a newly defined

where

It is evident that

lent to

On the other hand, we can verify directly that

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

Thus we have

where

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

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

which can be verified readily. When we concern about the non-iso-spectral case, (4.7) holds for

we first prove

Denote

It is ease to know “

and

Their difference is

The Equation (4.11) is proved.

On the other hand, through comparing the coefficients of

That is

Using the recursion relations of

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

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 [

Consider transformation

where

Lemma 5.1. (see also [

where

where

Proof. Transformation T as DT must solve the following equation

Comparing coefficients of

Suppose

Transformation (5.1) also change (2.1) to time evolution which matches to

where

We will prove that_{11}, V_{22} and V_{12}, V_{21} are polynomials of

and

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 _{11} is polynomial of

Substituting them and (5.4) into V_{11} gives rise to

It is not difficult to check that

is k + 1 power polynomial of_{11}, obviously,

as _{1}, h_{4} tend to

to V_{12}, 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

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

From gauge transformation

which forms DT of generalized Ragnisco-Tu equation. Matrix

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

When seed solution

These equalities produce

and

Thus we get

and form (5.17c), we obtain

Substituting it into (5.21),

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.

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

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