Common Hirota Form Bäcklund Transformation for the Unified Soliton System
aa r X i v : . [ n li n . S I] A p r Common Hirota Form B¨acklund Transformationfor the Unified Soliton System
Masahito Hayashi ∗ Osaka Institute of Technology, Osaka 535-8585, JapanKazuyasu Shigemoto † Tezukayama University, Nara 631-8501, JapanTakuya Tsukioka ‡ Bukkyo University, Kyoto 603-8301, Japan
Abstract
We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) ∼ = GL(2, R ) ∼ = M¨obius 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 B¨acklund transformation,we can transform different KdV/mKdV/sinh-Gordon equations and the B¨acklund transforma-tions of the standard form into the same common Hirota form and the same common B¨acklundtransformation except the equation which has the time-derivative term. The difference is onlythe time-dependence and the main structure of the N -soliton solutions has the same commonform for KdV/mKdV/sinh-Gordon systems. Then the N -soliton solutions for the sinh-Gordonequation is obtained just by the replacement from KdV/mKdV N -soliton solutions.We also give general addition formulae coming from the KdV type B¨acklund transformationwhich plays not only an important role to construct the trigonometric/hyperbolic N -solitonsolutions but also an essential role to construct the elliptic N -soliton solutions. In contrast tothe KdV type B¨acklund transformation, the well-known mKdV/sinh-Gordon type B¨acklundtransformation gives the non-cyclic symmetric N -soliton solutions. We give an explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equations. Studies of soliton systems have a long history. The discovery of the soliton system by theinverse scattering method [1–3] has given the breakthrough to exactly solve some specialnon-linear equations. There have been many interesting developments to understand solitonsystems such as the AKNS formulation [4, 5], the B¨acklund transformation [6–9], the Hirotaequation [9–13], the Sato theory [14], the vertex construction of the soliton solution [15,16], andthe Schwarzian type mKdV/KdV equation [17]. For the construction of N -soliton solutionsof various soliton equations, see the Wawzaz’s nice textbook [18]. Even now the solitontheory is quite actively studied in applying to the various non-linear phenomena such as ∗ [email protected] † [email protected] ‡ [email protected] ∼ = GL(2, R ) ∼ = M¨obius group point of view [23, 24]. We expectthat the various approaches above [1–17] are connected through the Lie group. We have alsoformulated soliton systems in a unified manner through the Einstein manifold of AdS in theRiemann geometry, which has SO(2,1) Lie group structure [25].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 systemati-cally from N pieces of 1-soliton solutions via algebraic addition formulae coming from theB¨acklund transformation. As a result, an expression of the N -soliton solutions becomes arational function of polynomial of many 1-soliton solutions. In the representation of theaddition formula of SO(2,1) ∼ = GL(2, R ) ∼ = M¨obius group, algebraic functions such as trigono-metric/hyperbolic/elliptic functions come out. We consider SO(2,1) ∼ = GL(2, R ) ∼ = M¨obiusgroup as the keystone for the soliton system. In the group theoretical point of view, wecan connect and unify various approaches for soliton systems. As the M¨obius group is therational transformation, it is natural to use rational Hirota variables. Furthermore, as theB¨acklund transformation can be considered as the self-gauge transformation, it is natural touse B¨acklund transformation as some addition formula of the M¨obius group in our Lie groupapproach.The B¨acklund transformation goes back to Bianchi [26] for the sine-Gordon equation. It isone of the strong tools to construct N -soliton solutions. For the old and recent developmentof the B¨acklund transformation, see the Rogers-Shadowick’s and the Rogers-Schief’s nicetextbooks [27, 28]. The recent hot topics of the B¨acklund transformation is the application ofB¨acklund transformation to the integrable defect [29–32].In this paper, N -soliton solutions would be categorized in terms of two types of the B¨ackludtransformation. We show one is the well-known KdV type B¨acklud transformation thatprovides cyclic symmetric N -soliton solutions, while another is the well-known mKdV/sinh-Gordon type B¨acklund transformation that gives non-cyclic symmetric solutions. We alsogive a general addition formula of the KdV type B¨acklund 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. The KdV equation is given by u t − u xxx + 6 uu x = 0 . (2.1)Introducing the τ -function by u = z x = − τ ) xx , the KdV equation becomes ∂∂x (cid:20) ( − D t D x + D x ) τ · ττ (cid:21) = 0 , (2.2)where D t , D x are Hirota derivatives defined by D kx f ( x ) · g ( x ) = f ( x )( ←− ∂ x − −→ ∂ x ) k g ( x ). Then theKdV equation turns to be so-called Hirota form( − D t D x + D x ) τ · τ = C τ , (2.3) In the representation of the addition formula of the SO(3) group, the elliptic function comes out [33, 34]. ith C as an integration constant. The C = 0 case corresponds to the elliptic soliton case. Here we take the special case i.e. C = 0 to consider only the trigonometric/hyperbolic solitonsolution, and we consider the special KdV equation in the form( − D t D x + D x ) τ · τ = 0 . (2.4)One soliton solution for this special Hirota type KdV equation is given by τ = 1 + e X i , with X i = a i x + a i t + c i . The Hirota type B¨acklund transformations in this case are given by( − D t + 3 a D x + D x ) τ ′ · τ = 0 , (2.5a) D x τ ′ · τ − a τ ′ τ = 0 . (2.5b)In fact, using the following relation [9], (cid:2) ( − D t D x + D x ) τ · τ (cid:3) τ ′ − τ (cid:2) ( − D t D x + D x ) τ ′ · τ ′ (cid:3) = − D x (cid:20)(cid:18) ( − D t + 3 a D x + D x ) τ ′ · τ (cid:19) · τ ′ τ + 3 (cid:0) D x τ ′ · τ (cid:1) · (cid:18) D x τ ′ · τ − a τ ′ τ (cid:19)(cid:21) , (2.6)we can show that if τ is the solution of Eq.(2.4) and if we use Eqs.(2.5a) and (2.5b) as theB¨acklund transformations, then τ ′ satisfies( − D t D x + D x ) τ ′ · τ ′ = 0 , (2.7)which means that τ ′ is a new solution.We now show that the Hirota type B¨acklund transformation Eq.(2.5b) relates to the fol-lowing well-known KdV type B¨acklund transformation z x + z ′ x = − a z − z ′ ) . (2.8)Writing down Eq.(2.5b) more explicitly, D x τ ′ · τ = τ ′ τ xx − τ ′ x τ x + τ ′ xx τ = a τ ′ τ, (2.9)and defining z = − τ x τ and z ′ = − τ ′ x τ ′ , we can organize Eq.(2.8) as z ′ x + z x + a −
12 ( z ′ − z ) = − τ ′ xx τ ′ + 2 τ ′ x τ ′ − τ xx τ + 2 τ x τ + a − (cid:18) τ ′ x τ ′ − τ x τ (cid:19) = − τ ′ τ (cid:18) τ ′ τ xx − τ ′ x τ x + τ ′ xx τ − a τ ′ τ (cid:19) = − τ ′ τ (cid:18) D x τ ′ · τ − a .τ ′ τ (cid:19) , (2.10)which leads the following equivalence D x τ ′ · τ = a τ ′ τ ⇐⇒ z ′ x + z x = − a z ′ − z ) . (2.11) In the static case, we take the τ -function as the Weierstrass’s σ -function, then D x τ · τ = C τ becomes ℘ xx = 6 ℘ − C /
2, which means that C = g in the standard notation. n the previous paper [23], we make the connection between the KdV equation and themKdV equation through the Miura transformation u = ± v x + v with the common Hirotatype variables f and g , that is, u = − τ ) xx , τ = f ± g in the KdV equation and v = w x ,tanh w/ g/f in the mKdV equation. In order to connect the KdV equation with the mKdVequation, we would like to take variables f and g as τ = f ± g , τ ′ = f ′ ± g ′ . For the N -solitonsolution, f and g are an even and an odd part of a N -soliton solution under changing anoverall sign of each 1-soliton solution. We refer f and g as Hirota form variables. In order toconstruct N -soliton solutions, only one of the B¨acklund transformations Eq.(2.5b) is enough,which is given by D x ( f ′ ± g ′ ) · ( f ± g ) = a f ′ ± g ′ )( f ± g ) . (2.12)We can simplify Eq.(2.4) by using f and g variables. By using the soliton number un-changing self B¨acklund transformation, i.e. f ′ = f , g ′ = − g , and a = 0 in Eq.(2.12), wehave D x ( f · f − g · g ) = 0 . (2.13)While by using p = f + g and q = f − g , we obtain an identity (cid:0) ( − D t D x + D x ) p · p (cid:1) q − p (cid:0) ( − D t D x + D x ) q · q (cid:1) = D x (cid:2) (cid:0) ( − D t + D x ) p · q (cid:1) · pq + 12 (cid:0) D x ( f · f − g · g ) (cid:1) · ( D x ( f · g )) (cid:3) . (2.14)Since we have ( − D t D x + D x ) p · p = 0 and ( − D t D x + D x ) q · q = 0 from Eq.(2.4) with τ = f ± g ,if we use Eq.(2.13), we have ( − D t + D x ) p · q = − − D t + D x )( f · g ) = 0. In this way, Eq.(2.4)is simplified in the following forms ( − D t + D x ) f · g = 0 , (2.15a) D x ( f · f − g · g ) = 0 . (2.15b)We call Eq.(2.15b) as a structure equation, which determines the structure of N -soliton so-lutions. While we refer Eq.(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 thosein the special mKdV equation. The mKdV equation is given by v t − v xxx + 6 v v x = 0 . (2.16)Defining v = w x and tanh( w/
2) = g/f , we get( − D t + D x ) f · gD x f · g = 3 D x ( f · f − g · g ) f − g . (2.17)We now consider the following special case( − D t + D x ) f · g = 0 , (2.18a) D x ( f · f − g · g ) = 0 . (2.18b)Then we have the common structure equation Eq.(2.18b) in the mKdV equation as that ofEq.(2.15b) in the KdV equation. Further we have the common dynamical equation Eq.(2.18a)in the mKdV equation as that of Eq.(2.15a) in the KdV equation. ne soliton solution for this special Hirota type mKdV equation (2.18a) and (2.18b) isgiven by f = 1 , g = e X i , with X i = a i x + a i t + c i . The B¨acklund transformation for the structure equation (2.18b) is given by [9] D x ( f ′ − g ′ ) · ( f + g ) = − a f ′ + g ′ )( f − g ) , (2.19a) D x ( f ′ + g ′ ) · ( f − g ) = − a f ′ − g ′ )( f + g ) , (2.19b)by using the following relations. Taking Eqs.(2.19a) and (2.19b) into account, we have arelation (cid:2) D x ( f ′ + g ′ ) · ( f ′ − g ′ ) (cid:3) ( f + g )( f − g ) − ( f ′ + g ′ )( f ′ − g ′ ) (cid:2) D x ( f + g ) · ( f − g ) (cid:3) = D x h(cid:16) D x ( f ′ + g ′ ) · ( f − g ) + a f ′ − g ′ )( f + g ) (cid:17) · ( f ′ − g ′ )( f + g ) − (cid:16) D x ( f ′ − g ′ ) · ( f + g ) + a f ′ + g ′ )( f − g ) (cid:17) · ( f ′ + g ′ )( f − g ) i + D x h − a f ′ − g ′ )( f + g ) · ( f ′ − g ′ )( f + g ) + a f ′ + g ′ )( f − g ) · ( f ′ + g ′ )( f − g ) i = D x h(cid:16) D x ( f ′ + g ′ ) · ( f − g ) + a f ′ − g ′ )( f + g ) (cid:17) · ( f ′ − g ′ )( f + g ) − (cid:16) D x ( f ′ − g ′ ) · ( f + g ) + a f ′ + g ′ )( f − g ) (cid:17) · ( f ′ + g ′ )( f − g ) i , (2.20)where we have used D x F · F = F x F − F F x = 0. This relation means that if Eqs.(2.18b),(2.19a), and (2.19b) are satisfied, we have D x ( f ′ · f ′ − g ′ · g ′ ) = 0, that is, if the set ( f, g ) is asolution, the set ( f ′ , g ′ ) produces a new solution by using the B¨acklund transformation.We can find equivalent forms for the B¨acklund transformations (2.19a) and (2.19b) [9].First, we consider the following relation D x ( f ′ + g ′ ) · ( f + g )( f ′ + g ′ )( f + g ) − D x ( f ′ − g ′ ) · ( f − g )( f − + g ′ )( f − g )= 1( f ′ − g ′ )( f − g ) D x h(cid:16) D x ( f ′ + g ′ ) · ( f − g ) + a f ′ − g ′ )( f + g ) (cid:17) · ( f ′ − g ′ )( f + g )+ ( f ′ + g ′ )( f − g ) · (cid:16) D x ( f ′ − g ′ ) · ( f + g ) + a f ′ + g ′ )( f − g ) (cid:17)i = 0 , (2.21)where we have used the B¨acklund transformations (2.19a) and (2.19b). Secondly, we obtain D x ( f ′ + g ′ ) · ( f + g )( f ′ + g ′ )( f + g ) + D x ( f ′ − g ′ ) · ( f − g )( f − g ′ )( f − g ) − a (cid:20) D x ( f ′ · f ′ − g ′ · g ′ )( f ′ − g ′ ) + D x ( f · f − g · g )( f − g ) (cid:21) + 2 (cid:20)(cid:18) D x ( f ′ + g ′ ) · ( f − g )( f ′ − g ′ )( f + g ) (cid:19) (cid:18) D x ( f ′ − g ′ ) · ( f + g )( f ′ + g ′ )( f − g ) (cid:19) − a (cid:21) = 2 (cid:20)(cid:18) D x ( f ′ + g ′ ) · ( f − g )( f ′ − g ′ )( f + g ) (cid:19) (cid:18) D x ( f ′ − g ′ ) · ( f + g )( f ′ + g ′ )( f − g ) (cid:19) − a (cid:21) = 0 , (2.22)where we have used the structure equations D x ( f ′ · f ′ − g ′ · g ′ ) = 0 and D x ( f · f − g · g ) = 0and also the B¨acklund transformations (2.19a) and (2.19b). Combining Eqs.(2.21) and (2.22),we arrive at D x ( f ′ ± g ′ ) · ( f ± g ) = a f ′ ± g ′ )( f ± g ) . (2.23) hen we have the common Hirota form B¨acklund transformation Eq.(2.23) in the mKdVequation as that of Eq.(2.12) in the KdV equation. This is the reason why we call this as thecommon KdV type Hirota form B¨acklund transformation.Conversely, if Eq.(2.23) is satisfied, we have D x h(cid:16) D x ( f ′ + g ′ ) · ( f − g ) + a f ′ − g ′ )( f + g ) (cid:17) · ( f ′ − g ′ )( f + g )+( f ′ + g ′ )( f − g ) · (cid:16) D x ( f ′ − g ′ ) · ( f + g ) + a f ′ + g ′ )( f − g ) (cid:17)i = 0 , (2.24) (cid:18) D x ( f ′ + g ′ ) · ( f − g )( f ′ − g ′ )( f + g ) (cid:19) (cid:18) D x ( f ′ − g ′ ) · ( f + g )( f ′ + g ′ )( f − g ) (cid:19) − a , (2.25)which give Eq.(2.19a) and Eq.(2.19b) by properly choosing the sign of a . Then we concludethe equivalence D x ( f ′ ± g ′ ) · ( f ∓ g )( f ′ ∓ g ′ )( f ± g ) = − a ⇐⇒ D x ( f ′ ± g ′ ) · ( f ± g )( f ′ ± g ′ )( f ± g ) = a . (2.26)The Eq.(2.23) is the Hirota type B¨acklund transformation for the special mKdV structureequation Eq.(2.18b).Now we focus on yet another mKdV type B¨acklund transformation [9] w ′ x + w x = a sinh( w ′ − w ) . (2.27)From Eqs.(2.19a) and (2.19b), we can obtain (2.27), while the opposite is not always true: D x ( f ′ ± g ′ ) · ( f ∓ g ) = − a f ′ ∓ g ′ )( f ± g ) = ⇒ w ′ x + w x = a sinh( w ′ − w ) . (2.28)We can show the relation above in the following manner. Usingtanh w gf , sinh w = 2 f gf − g , cosh w = f + g f − g , and their counterparts for ( w ′ , f ′ , g ′ ), we have w x = 2( f g x − f x g ) f − g = − D x ( f − g ) · ( f + g ) f − g , and those for ( w ′ , f ′ , g ′ ). Then we have a relation h D x ( f ′ + g ′ ) · ( f − g ) + a f ′ − g ′ )( f + g ) i ( f ′ − g ′ )( f + g ) − ( f ′ + g ′ )( f − g ) h D x ( f ′ − g ′ ) · ( f + g ) + a f ′ + g ′ )( f − g ) i = (cid:2) D x ( f ′ + g ′ ) · ( f ′ − g ′ ) (cid:3) ( f − g )( f + g ) − ( f ′ + g ′ )( f ′ − g ′ ) [ D x ( f − g ) · ( f − g )]+ a (cid:2) ( f ′ − g ′ ) ( f + g ) − ( f ′ + g ′ ) ( f − g ) (cid:3) = ( f ′ − g ′ )( f − g ) (cid:20) w ′ x + w x + a (cid:18) f ′ + g ′ f ′ − g ′ f gf − g − f ′ g ′ f ′ − g ′ f + g f − g (cid:19)(cid:21) = ( f ′ − g ′ )( f − g ) (cid:2) w ′ x + w x + a (cosh( w ′ ) sinh( w ) − sinh( w ′ ) cosh( w ) (cid:3) = ( f ′ − g ′ )( f − g ) (cid:2) w ′ x + w x − a sinh( w ′ − w ) (cid:3) , (2.29)which means we have Eq.(2.27) from Eqs.(2.19a) and (2.19b), but the opposite is not alwaysshown. In fact, Eq.(2.27) is the B¨acklund transformation of the original mKdV equation q.(2.17) but not the B¨acklund transformation of the special mKdV equations Eqs.(2.18a)and (2.18b).By the KdV type Hirota form B¨acklund transformation Eq.(2.23), we have the cyclic sym-metric N -soliton solutions. On the other hand, by the mKdV type B¨acklund transformationEq.(2.27), we have the non-cyclic symmetric N -soliton solutions. In section 4, we give anexplicit non-cyclic symmetric 3-soliton solution from mKdV type B¨acklund transformationEq.(2.27). The sinh-Gordon equation is given by θ xt = sinh θ. (2.30)Defining tanh( θ/
4) = g/f , we obtain D t D x f · gf g − D t D x ( f · f + g · g ) f + g . (2.31)We here consider the special case: D t D x f · g = f g, (2.32a) D t D x ( f · f + g · g ) = 0 . (2.32b)Taking the following relation into account, D x (cid:2) D t D x ( f · f + g · g ) · ( f + g ) − D t D x ( f · g ) − f g ) · f g (cid:3) = D t (cid:2)(cid:0) D x ( f · f − g · g ) (cid:1) · ( f − g ) (cid:3) , (2.33)we take D t D x f · g = f g, (2.34a) D x ( f · f − g · g ) = 0 , (2.34b)as the special sinh-Gordon equation instead of Eqs.(2.32a) and (2.32b). The above structureequation Eq.(2.34b) in the sinh-Gordon equation is the same as that of Eq.(2.15b) in the KdVequation and Eq.(2.18b) in the mKdV equation. Then, applying the same method as that ofthe mKdV equation, we have the common KdV type Hirota form B¨acklund transformationEqs.(2.19a) and (2.19b) , and equivalently Eq.(2.23) for KdV/mKdV/sinh-Gordon equations.One soliton solution for this special type sinh-Gordon equation is given by f = 1 , g = e ˆ X i , with ˆ X i = a i x + t/a i + c i . From Eqs.(2.19a) and (2.19b), we have another mKdV type B¨acklund transformation byreplacing w → θ/ w/
2) = g/f in the mKdV equation corresponds to tanh( θ/
4) = g/f in thesinh-Gordon equation. Then from Eqs.(2.19a) and (2.19b), we have θ ′ x θ x a sinh (cid:18) θ ′ − θ (cid:19) , (2.35)but the opposite is not always satisfied. In fact, Eq.(2.35) is the B¨acklund transformationfor the original sinh-Gordon equation Eq.(2.31) but not the B¨acklund transformation of thespecial sinh-Gordon equation Eqs.(2.34a) and (2.34b). .4 Cyclic symmetric N -soliton solutions via Hirota formB¨acklund transformations Let us first summarize our findings in the previous subsections. By using the Hirota formvariables f and g , we can treat the special KdV/mKdV/sinh-Gordon equations in a unifiedmanner: a) KdV Eq. : u = z x = − τ ) xx , τ = f ± g, (2.36)b) mKdV Eq. : v = w x , tanh w gf , (2.37)c) sinh-Gordon Eq. : tanh θ gf . (2.38)The well-known KdV type B¨acklund transformation is equivalent to the KdV type Hirotaform B¨acklund transformation: z ′ x + z x = − a z ′ − z ) ⇐⇒ D x ( f ′ ± g ′ ) · ( f ± g ) = a f ′ ± g ′ ) · ( f ± g ) . (2.39)We have the common KdV type Hirota form B¨acklund transformation Eq.(2.39) for the spe-cial KdV equation Eq.(2.15a) and Eq.(2.15b), for the special mKdV equation Eqs.(2.18a) and(2.18b), and for the special sinh-Gordon equation Eqs.(2.34a) and (2.34b) for the commonstructure equation Eqs.(2.15b), (2.18b) and (2.34b). Another mKdV type B¨acklund transfor-mation Eq.(2.27) is the B¨acklund transformation of the original mKdV equation Eq.(2.17) butnot the B¨acklund transformation of the special mKdV equation Eq.(2.18a) and Eq.(2.18b).In our previous paper [23], we have demonstrated how to construct N -soliton solutionsfrom N pieces of 1-soliton solutions by using KdV type B¨acklund transformation Eq.(2.8).Here we demonstrate how to construct the cyclic N -soliton solutions for N =2 case. We startfrom the addition formula of the B¨acklund transformation, z = a − a z − z , (2.40)where we choose z = 0 , z i = − a i tanh X i / , with X i = a i x + a i t + c i . In order to find a KdV two-soliton solution, we simply take the space derivative by using u = z ,x . While, if we want to find a 2-soliton solution for the mKdV/sinh-Gordon equation, wemust know f and g from z . We can find f and g from z = − τ ,x /τ + const. with τ = f ± g [23], but it becomes complicated for the general N -soliton solutions. However,it is easier to find the τ -function directly from the Hirota equation ( − D t D x + D x ) τ · τ = 0in the standard way [13, 18], which gives τ = f ± g (2.41)with f = 1 + ( a − a ) ( a + a ) e X e X , (2.42a) g = e X + e X , (2.42b)where f and g are even and odd parts of the τ function under e X i → − e X i . For a 2-solitonsolution of mKdV equation, we have tanh( w /
2) = g /f [18]. For a soliton solution of inh-Gordon equation, using the dynamical equation Eq.(2.34a), we replace X i → ˆ X i withˆ X i = a i x + t/a i + c i , because f = 1, g = e ˆ X i is a 1-soliton solution of D t D x f · g = f g .Then the 2-soliton solution of sinh-Gordon equation is given by tanh θ / g/ ˆ f [18], whereˆ f = 1 + ( a − a ) / ( a + a ) e ˆ X e ˆ X , ˆ g = e ˆ X + e ˆ X .In general, we have the cyclic symmetric N -soliton solutions [18] by using the commonKdV type B¨acklund transformation. In our approach, we construct cyclic symmetric N -soliton solutions by an algebraic additionformula coming from the well-known KdV type B¨acklund transformation, which is equivalentto the common KdV type B¨acklund transformation. This addition formula is applicablealso to construct the elliptic N -soliton solutions and there will be no other way to construct N -soliton solutions for the elliptic case [24]. In order to construct N -soliton solutions fortrigonometric/hyperbolic/elliptic soliton solutions, we give the result of the general additionformula here.Let us first review to find a 2-soliton solution by the common KdV type B¨acklund trans-formation. Assuming the commutativity, z = z , we have z ,x + z ,x = − a z − z ) , (3.1a) z ,x + z ,x = − a z − z ) , (3.1b) z ,x + z ,x = − a z − z ) , (3.1c) z ,x + z ,x = − a z − z ) . (3.1d)Making Eq.(3.1a) − Eq.(3.1b) − Eq.(3.1c)+Eq.(3.1d), derivative terms are canceled out and wehave z = z + a − a z − z . (3.2)We can check that Eq.(3.2) satisfies Eqs.(3.1a)-(3.1d), which means that it is commutative inthis level. Recursively, we have z ··· ,n − ,n − ,n = z ··· ,n − + a n − − a n z ··· ,n − ,n − − z ··· ,n − ,n . (3.3)We list various N -soliton solutions obtained through the addition formulae: • (2+1)-soliton solution z = z + a − a z − z = z + G F , (3.4)with F = z − z , (3.5a) G = a − a . (3.5b) z = z + a − a z − z = G F , (3.6)with F = ( a − a ) z + ( a − a ) z + ( a − a ) z = 12! X i,j,k =1 ǫ ijk ( a i − a j ) z k , (3.7a) G = − (cid:0) ( a − a ) z z + ( a − a ) z z + ( a − a ) z z (cid:1) = − X i,j,k =1 ǫ ijk ( a i − a j ) z i z j . (3.7b) • (4+1)-soliton solution z = z + a − a z − z = z + G F , (3.8)with F = 1(2!) X i,j,k,l =1 ǫ ijkl ( a i − a j )( a k − a l ) z i z j , (3.9a) G = − X i,j,k,l =1 ǫ ijkl a i a j ( a i − a j ) z k . (3.9b) • z = z + a − a z − z = G F , (3.10)with F = 13!2! X i,j,k,l,m =1 ǫ ijklm ( a i − a j ) (cid:2) ( a k − a l )( a l − a m )( a m − a k ) (cid:3) z i z j , (3.11a) G = 13!2! X i,j,k,l,m =1 ǫ ijklm ( a i − a j ) (cid:2) ( a k − a l )( a l − a m )( a m − a k ) (cid:3) z k z l z m , (3.11b)where ǫ i i ··· i n is a Levi-Civita symbol with ǫ ··· n = 1. We first define the following quantityΛ( i , i , · · · , i n ) = n X p,q =1 p 2) tanh( w / , (4.5)tanh (cid:16) w (cid:17) = − a tanh (cid:16) w − w (cid:17) = − a tanh( w / − tanh( w / − tanh( w / 2) tanh( w / , (4.6)with a ij = ( a i − a j ) / ( a i + a j ) = − a ji .Next, let us construct a 3-soliton solution. Assuming the commutativity w = w , wehave tanh (cid:16) w − w (cid:17) = − a + a a − a tanh (cid:16) w − w (cid:17) . (4.7)We express the above with t = tanh( w / t = tanh( w / t = tanh( w / (cid:16) w (cid:17) = g f , with f = X i =0 c i p i ( t ) , g = X i =0 c i q i ( t ) . (4.8)In the expression above, we denote c = a a a , c = − a + a − a , c = − a a a − a + a ,c = − a a a + a + a , c = − a , c = a , c = − a , c = a a a − a + a ,p = 1 , p = t , p = t t , p = t t , p = t t , p = t t , p = t t , p = t t t ,q = t t t , q = t t t , q = t t , q = t t , q = t , q = t , q = t , q = 1 , which satisfy p i q i = t t t ( i = 0 , , · · · , w / 2) is not cyclicsymmetric in t , t , and t . This is the non-cyclic symmetric 3-soliton solution of the mKdVequation derived from another mKdV type B¨acklund transformation.The non-cyclic symmetric 3-soliton solution for the sinh-Gordon equation can be obtainedby replacing tanh( w / → tanh( θ / 4) and t i = tanh w i / .X i → ˆ t i = tanh θ i / ˆ X i .We can connect the mKdV equation with the sinh-Gordon equation in another way. If weput w = c in Eq.(4.2), we have w ′ x = a sinh( w ′ − c ) and w ′ t = a sinh( w ′ − c ), which givesthe sinh-Gordon equation Θ xt = a sinh(Θ) through the relation Θ = 2( w ′ − c ), and the a -dependence can be eliminated by the redefinition of x → x/a , and t → t/a . Summary and discussions We consider the reason why special non-linear differential equations, such as KdV/mKdV/sinh-Gordon equations, have the systematic N -soliton solution is because such soliton equationshave SO(2,1) ∼ = GL(2, R ) ∼ = M¨obius group structure. The systematic N -soliton solutionsare given as the result of the addition formula of these Lie groups. As the representa-tion of the addition formula of the Lie groups, the algebraic function such as trigonomet-ric/hyperbolic/elliptic functions appear.We have studied to unify the soliton system through the common addition formula com-ing from the common KdV type Hirota form B¨acklund transformation D x ( f ′ ± g ′ ) · ( f ± g ) = a ( f ′ ± g ′ )( f ± g ) / 4, which is equivalent to the well-known KdV type B¨acklund transformation z ′ x + z x = − a / z ′ − z ) / z = − f ± g )] x , z ′ = − f ′ ± g ′ )] x . If we constructthe N -soliton solutions through the KdV type B¨acklund transformation, we can transformdifferent KdV/mKdV/sinh-Gordon equations and B¨acklund transformations of the standardform into the same common Hirota form and B¨acklund transformation, Eq.(2.12), Eq.(2.15b),Eq.(2.23), Eq,(2.18b) and Eq.(2.34b) except the equation which has the time-derivative term.In KdV/mKdV equation, the equation which has the time-derivative term becomes the sameEq.(2.15a) and Eq.(2.18a) but it is different from sinh-Gordon’s one Eq.(2.34a). The differ-ence is only the time-dependence and the main structure of the N -soliton solutions has thesame common form for KdV/mKdV/sinh-Gordon systems. Then the N -soliton solutions forthe sinh-Gordon equation is obtained just by the replacement a i x + a i t → a i x + t/a i fromKdV/mKdV N -soliton solutions.We have also given the general addition formula of this common KdV type Hirota formB¨acklund transformation. This addition formula is applicable also to construct the ellip-tic N -soliton solutions and there will be no other way to construct N -soliton solutions forthe elliptic case [24]. Then it is useful to construct N -soliton solutions for trigonomet-ric/hyperbolic/elliptic soliton solutions.While by using another mKdV/sinh-Gordon type B¨acklund transformation w ′ x + w x = a sinh( w ′ − w ), we have the non-cyclic symmetric solution. For the non-cyclic symmetric N -soliton solutions for the KdV equation, we can construct that from the mKdV non-cyclicsymmetric N -soliton solutions through the Miura transformation u = ± v x + v . 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