The Unified Soliton System as the ${\rm AdS_2}$ System

We study the Riemann geometric approach to be aimed at unifying soliton systems. The general two-dimensional Einstein equation with constant scalar curvature becomes an integrable differential equation. We show that such Einstein equation includes KdV/mKdV/sine-Gordon equations.

Our characterization of the soliton system is that it is a system of the integrable non-linear differential equation, which has not only some exact solutions but also N -soliton solutions (systematic infinitely many solutions). The KdV/mKdV/sine-Gordon systems are such soliton systems, and we will study the common structure for unified soliton systems.
In our previous paper [16,17], we have characterized such soliton system as it has the local GL(2,R) self gauge symmetry, where the local gauge parameter is connected with the gauge potential. We have pointed out that a special local self gauge transformation becomes the Bäcklund transformation. Combining various Bäcklund transformations, we have the algebraic relation, which becomes the addition formula to construct the N -soliton solution from various 1-soliton solutions. In our approach, the mechanism to be able to construct Nsoliton solutions for the non-linear soliton system comes from the GL(2,R) group (=Möbius group) structure for such system.
So far almost all soliton system, which admits N -soliton solution, is restricted only to the two-dimensional model, and the Lie group structure of such soliton system is restricted to the rank one Lie group GL(2,R)/Z 2 ∼ = SO(2, 1) ∼ = SU(1, 1)/Z 2 . Then we will study in this paper the reason why the usual soliton system is restricted to the two-dimensional model and has the rank one Lie group structure from the Riemann geometric approach.
The sine-Gordon equation is the well-known integrable system from old days [20]. In the theory of the curved surface, the fundamental equation is known as the Gauss-Weingarten equation. The integrability condition of this Gauss-Weingarten equation is known as the Gauss-Codazzi formula. The sine-Gordon equation comes from this Gauss-Codazzi formula for the pseudo-sphere. However, the approach from the curved surface is more complicated than the approach from the Riemann geometry. The approach from the curved surface seems to be difficult to generalize it into the higher dimensional and higher Lie group symmetric soliton system. Then we use the Riemann geometric approach to the soliton system in this paper.
We will show in this paper, i) the general two-dimensional Einstein equation with constant negative scalar curvature, that is AdS 2 system, becomes the integrable differential equation. ii) such Einstein equation includes the unified soliton system of the KdV/mKdV/sine-Gordon equations.
2 Riemann geometric approach to two-dimensional soliton system as AdS 2 system We start from the general two-dimensional metric in the form In two dimensions, due to the first Bianchi identity of the Riemann tensor, we have We consider the case of the constant scalar curvature R, that is, R = 2K(x, t) = constant, and we take K(x, t) = −1 for simplicity. In this case, the metric becomes the Einstein metric R ij = −g ij . Then we have R = −2, which gives the following differential equation Under the general coordinate transformation in two dimensions, t → T (t, x) and x → X(t, x), the Einstein metric R ij = −g ij can be transformed into the metric of the pseudosphere x 2 + y 2 − z 2 = −1, which has the symmetry of SO(2,1). We parametrize it in the form x = sinh θ cos φ, y = sinh θ sin φ, z = cosh θ, which gives dx = cosh θ cos φ dθ − sinh θ sin θ dφ, dy = cosh θ sin φ dθ + sinh θ cos θ dφ, These lead the metric which is one of the AdS 2 parametrizations. For this metric, we have In general, the Lie group symmetry of the general two-dimensional surface with the negative constant scalar curvature is that of the rank one Lie group GL(2,R)/Z 2 ∼ = SO(2, 1) ∼ = S(1, 1)/Z 2 .
In the following section, we show that the above differential equation is, i) the integrable differential equation, ii) it includes KdV, mKdV and sine-Gordon equations.
3 Integrable Condition of Surface in three-dimensional Euclidean Space

Point in three-dimensional Euclidean space
We formulate three-dimensional geometry by the Maurer-Cartan formalism with the exterior differential form [18,19]. For any point x, we attach the moving orthonormal basis e 1 , e 2 , e 3 . The differential 1-form dx leads the structure equation given in the form The torsion free condition is defined by By using the first Bianchi identity, the general solution of Eq.(3.4) is given by where (R) ijkl =R ijkl is the Riemann tensor. This is called as the curvature condition.

Point on surface in three-dimensional Euclidean space
We formulate the unified two-dimensional soliton system as the system of the negative constant surface in this Maurer-Cartan geometry [5,6]. We choose the normal vector of the surface as e 3 . Then the point on the surface is given by σ 3 = 0. Further, we have ω 1 = 0, ω 2 = 0 from Eq. (3.3). The structure equation becomes dx = σ 1 e 1 + σ 2 e 2 , (3.6) The integrability condition turns to be where we denote K = −R 0101 as the Gauss curvature. Here we consider the pseudo-sphere, that is, the constant negative curvature and we take K = −1 for simplicity. Then we have the integrability condition in the form (3.14) This integrability condition can be expressed in the Sasaki's 2 × 2 matrix form [6] dΩ =Ω ∧Ω, withΩ = 1 2

General integrable differential equation
We start from the general 1-forms σ 1 , σ 2 , and we construct ω 3 from Eq.(3.12) and Eq.(3.13). Then Eq.(3.14) gives the integrable differential equation. We put From Eq.(3.12) and Eq.(3.13), we have Then Eq.(3.14) gives the following integrable differential equation in the form ∂ ∂x In this case, the metric is given by where 4 KdV/mKdV/sine-Gordon equations as AdS 2 differential equation 4

KdV equation
For the AKNS formalism of the KdV equation we use the Sasaki's form [6] to obtain the simple expression of the integrable geometrical differential equation. Then we take dΩ =Ω ∧Ω, Denoting P = u t + 6uu x + u xxx , Q = P x = u xt + 6uu xx + 6u 2 x + u xxxx , and after calculating the scalar curvature, we have Then if the KdV equation Eq.(4.9) is satisfied, we have P = 0 and Q = 0, which gives R/2 = K = −1. However, the opposite is not always true, that is, even if R/2 = K = −1 is satisfied, we have the more general differential equation than the KdV equation and it contains the KdV equation as the special case. The reason of this property will come from the fact that Eq.(3.12) is not identically satisfied but gives the KdV equation itself. In the non-linear system, it is generally difficult to have the one-to-one correspondence between two integrable systems. For example, KdV system and mKdV system is connected by the Miura transformation u = ±iv x + v 2 , which means that if v satisfies the mKdV equation, u satisfies the KdV equation. But the opposite is not always true. That is, if u satisfies the KdV equation, v satisfies the more general differential equation than the mKdV equation and it contains the mKdV equation as the special case. In this KdV case, the integrability conditions Eqs.(3.12)-(3.14) are not in the standard form, then we have the more general differential equation than the KdV equation. After the two dimensional general coordinate transformation, we will be able to make the integrability conditions of the KdV equation in the standard form.

Summary and discussions
We have studied the Riemann geometric approach to the unified soliton systems, KdV/mKdV/ sine-Gordon equations. We have found that the general two-dimensional Einstein equation with constant negative scalar curvature becomes the integrable differential equation. Furthermore, we have explicitly shown that such Einstein equation includes KdV/mKdV/sine-Gordon equations.
The reason why the two-dimensional Einstein equation with negative constant scalar curvature becomes the integrable equation might be that such Einstein equation has the Einstein metric. Even in the higher dimensional and higher Lie group symmetric soliton equations, the existence of the Einstein metric may be essential. Then the Einstein equation for the AdS n =SO(2,n − 1)/SO(1,n − 1) system and/or the Einstein equation for the Kähler-Einstein manifold, both of these systems have the Einstein metric, may be the candidate of the higher dimensional soliton system. In two dimensions, the Einstein equation for the Kähler-Einstein manifold becomes the Liouville equation which is the integrable differential equation [21].