Common Hirota form Bäcklund transformation for the unified Soliton system

We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) ≅ GL(2, R ) ≅ Möbius group point of view, which might be a keystone to exactly solve some special non-linear differential equations. If we construct the N-soliton solutions through the KdV type Bäcklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the Bäcklund transformations of the standard form into the same common Hirota form and the same common Bäcklund transformation except the equation which has the time-derivative term. The difference is only the time-dependence and the main structure of the N-soliton solutions has the same common form for KdV/mKdV/sinh-Gordon systems. Then the N-soliton solutions for the sinh-Gordon equation is obtained just by the replacement from KdV/mKdV N-soliton solutions. We also give general addition formulae coming from the KdV type Bäcklund transformation which plays not only an important role to construct the trigonometric/hyperbolic N-soliton solutions but also an essential role to construct the elliptic N-soliton solutions. In contrast to the KdV type Bäcklund transformation, the well-known mKdV/sinh-Gordon type Bäcklund transformation gives the non-cyclic symmetric N-soliton solutions. We give an explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equations.

We refer a soliton system as that for special types of non-linear differential equations, which have not only exact solutions but also N-soliton solutions constructed systematically from N pieces of 1-soliton solutions via algebraic addition formulae coming from the Bäcklund transformation. As a result, an expression of the Nsoliton solutions becomes a rational function of polynomial of many 1-soliton solutions. In the representation of the addition formula of SO(2,1)≅GL(2, )≅Möbius group, algebraic functions such as trigonometric/ hyperbolic/elliptic functions 5 come out. We consider SO(2,1)≅GL(2, )≅Möbius group as the keystone for the soliton system. In the group theoretical point of view, we can connect and unify various approaches for soliton systems. As the Möbius group is the rational transformation, it is natural to use rational Hirota variables. Furthermore, as the Bäcklund transformation can be considered as the self-gauge transformation, it is natural to use Bäcklund transformation as some addition formula of the Möbius group in our Lie group approach.
The Bäcklund transformation goes back to Bianchi [28] for the sine-Gordon equation. It is one of the strong tools to construct N-soliton solutions. For the old and recent development of the Bäcklund transformation, see the Rogers-Shadowick's and the Rogers-Schief's nice textbooks [29,30]. The recent hot topics of the Bäcklund transformation is the application of Bäcklund transformation to the integrable defect [31][32][33][34].
In this paper, N-soliton solutions would be categorized in terms of two types of the Bäcklud transformation. We show one is the well-known KdV type Bäcklud transformation that provides cyclic symmetric N-soliton solutions, while another is the well-known mKdV/sinh-Gordon type Bäcklund transformation that gives noncyclic symmetric solutions. We also give a general addition formula of the KdV type Bäcklund transformation. An explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equation would be exposed. We are interested in the mathematical structure of the integrable soliton system, which has N-soliton solutions, we did not mention the physical applications in this paper.

KdV equation
The KdV equation is given by Introducing the τ-function by . Then the KdV equation turns to be so-called Hirota form with C 1 as an integration constant. The ¹ C 0 1 case corresponds to the elliptic soliton case. 6 Here we take the special case i.e. = C 0 1 to consider only the trigonometric/hyperbolic soliton solution, and we consider the special KdV equation in the form One soliton solution for this special Hirota type KdV equation is given by The Hirota type Bäcklund transformations in this case are given by In fact, using the following relation [9], we can show that if τ is the solution of equation (2.4) and if we use equations (2.5a) and (2.5b) as the Bäcklund transformations, then t¢ satisfies 5 In the representation of the addition formula of the SO(3) group, the elliptic function comes out [26,27]. 6 In the static case, we take the τ-function as the Weierstrass's σ-function, , which means that C 1 =g 2 in the standard notation.  Writing down equation (2.5b) more explicitly, which leads the following equivalence In the previous paper [23], we make the connection between the KdV equation and the mKdV equation with the common Hirota type variables f and g, that is, in the mKdV equation. In order to connect the KdV equation with the mKdV equation, we would like to take variables f and g as . For the N-soliton solution, f and g are an even and an odd part of a N-soliton solution under changing an overall sign of each 1-soliton solution. We refer f and g as Hirota form variables. In order to construct N-soliton solutions, only one of the Bäcklund transformations equation (2.5b) is enough, which is given by We can simplify equation (2.4) by using f and g variables. By using the soliton number unchanging self Bäcklund transformation, i.e. ¢ = ¢ =f f g g , , and a=0 in equation (2.12), we have While by using = + p f g and =q f g , we obtain an identity . In this way, equation (2.4) is simplified in the following forms We call equation (2.15b) as a structure equation, which determines the structure of N-soliton solutions. While we refer equation (2.15a) as a dynamical equation, which yields time dependence of N-soliton solutions. In next subsection, we will see that these equations are the same as those in the special mKdV equation.

mKdV equation
The mKdV equation is given by Defining v=w x and We now consider the following special case 2.18 The Bäcklund transformation for the structure equation (2.18b) is given by [9] ( 2 . 1 9 x by using the following relations. Taking equations (2.19a) and (2.19b) into account, we have a relation where we have used . This relation means that if equations (2.18b), (2.19a), and , that is, if the set ( f,g) is a solution, the set ( ) ¢ ¢ f g , produces a new solution by using the Bäcklund transformation.
We can find equivalent forms for the Bäcklund transformations (2.19a) and (2.19b) [9]. First, we consider the following relation where we have used the Bäcklund transformations (2.19a) and (2.19b). Secondly, we obtain and also the Bäcklund transformations (2.19a) and (2.19b which give equation (2.19a) and equation (2.19b) by properly choosing the sign of a. Then we conclude the equivalence The equation ( We can show the relation above in the following manner. Using and their counterparts for ( ) ¢ ¢ ¢ w f g , , , we have and those for ( ) ¢ ¢ ¢ w f g , , . Then we have a relation

sinh-Gordon equation
The sinh-Gordon equation is given by , we obtain We here consider the special case: Taking the following relation into account, we take 2.34

Cyclic symmetric N-soliton solutions via Hirota form Bäcklund transformations
Let us first summarize our findings in the previous subsections. By using the Hirota form variables f and g, we can treat the special KdV/mKdV/sinh-Gordon equations in a unified manner: The well-known KdV type Bäcklund transformation is equivalent to the KdV type Hirota form Bäcklund transformation: In our previous paper [23], we have demonstrated how to construct N-soliton solutions from N pieces of 1-soliton solutions by using KdV type Bäcklund transformation equation (2.8). Here we demonstrate how to construct the cyclic N-soliton solutions for N=2 case. We start from the addition formula of the Bäcklund transformation, where we choose In order to find a KdV two-soliton solution, we simply take the space derivative by using = u z x 12 12, . While, if we want to find a 2-soliton solution for the mKdV/sinh-Gordon equation, we must know f 12 and g 12 from z 12 . We can find f 12 and g 12 from t t = -+ z 2 c o n s t .

General formula
We first define the following quantity where we set Λ(i 1 , i 2 )=1. The general formula is expected to be given in the following form: • ((2n)+1)-solution This equation satisfies equations (4.3a)-(4.3d), so that w 12 can be new solution. Notice that from the timedependent 1-soliton solutions w 0 , w 1 , and w 2 , we obtain the time-dependent new solution w 12 , so that equation (4.2) is not necessary to construct the new solution. By using the above Bäcklund transformation, we can construct a new soliton solution w 12 from 1-soliton solutions w 1 ,w 2 , and w 0 . Taking that = w 0 0 is a trivial solution into account, we have 2-soliton solutions w 12 , and w 13 by using 1-soliton solutions w 1 , w 2 , and w 3 through