On Differential Equations Describing 3-Dimensional Hyperbolic Spaces

Transcription

1 Commun. Theor. Phys. Beijing, China pp c International Academic Publishers Vol. 45, No. 1, January 15, 26 On Differential Equations Describing 3-Dimensional Hyperbolic Spaces WU Jun-Yi, 1 DING Qing, 1,2 Keti Tenenblat 3 1 Institute of Mathematics, Fudan University, Shanghai 2433, China 2 Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai 2433, China 3 Department of Mathematics, Brasilia University, Brasilia DF 791-9, Brazil Received March 3, 25; Revised June 21, 25 Abstract In this paper, we introduce the notion of a 21-dimensional differential equation describing threedimensional hyperbolic spaces 3-h.s.. The 21-dimensional coupled nonlinear Schrödinger equation its sister equation, the 21-dimensional coupled derivative nonlinear Schrödinger equation, are shown to describe 3-h.s. The 21-dimensional generalized HF model: S t = 1 [S, Sy] 2iσSx, 2i σx = 1 trssxsy, in which S GL C 2, 4i GL C 1 GL C 1 provides another example of 21-dimensional differential equations describing 3-h.s. As a direct consequence, the geometric construction of an infinite number of conservation laws of such equations is illustrated. Furthermore we display a new infinite number of conservation laws of the 21-dimensional nonlinear Schrödinger equation the 21-dimensional derivative nonlinear Schrödinger equation by a geometric way. PACS numbers: 5.45.Yv, 2.4.Ky Key words: 21-dimensional integrable systems, differential equations describing 3-dimensional hyperbolic spaces, conservation laws 1 Introduction In 1979, Sasaki [1] observed that KdV, mkdv, SG in AKNS hierarchies see Ref. [2] are related to pseudospherical surfaces. The geometric notion of a 11- dimensional evolution equation describing pseudospherical surfaces p.s.s. was actually introduced by Chern Tenenblat in 1986 in Ref. [3], in which evolution equations of type: u t = F u, u x,..., u k x were systematically studied. In 1986, C.H. Gu H.S. Hu studied the similar problem in Ref. [4]. Later, in Refs. [5], [6], [7], this concept was applied to other types of differential equations. Geometric interpretation of some dynamical properties such as conservation laws, symmetries Bäcklund transformations of such equations have been explored in Refs. [1], [3], [5], [7], [8]. In 1995, Kamran Tenenblat gave a complete classification of evolution describing p.s.s. in Ref. [5]. The geometric notion of differential systems describing surfaces of constant nonzero Gaussian curvature was introduced by Ding Tenenblat in Ref. [9], in which they gave a characterization of evolution systems describing surfaces of constant nonzero Gaussian curvature classified the nonlinear Schrödinger type of differential systems describing an η-pseudospherical or η-spherical surface. It is well known that certain two-dimensional in spatial generalizations of the 11-dimensional integrable systems are related to physically interesting nonlinear equations of type u t = F u, u x,..., u k x evolution systems in 21-dimensions. The nonlinear Schrödinger equation NLS: iq t q xx 2κ q 2 q =, in which κ is a real constant, is a representative example in the theory of integrable systems, for the NLS is fundamental in many technical aspects illustrates directly some other integrable systems, such as KdV, mdv, SG, etc. see for example Ref. [1]. The NLS also models a wide range of physical phenomena see for example Ref. [7]. The following 21-dimensional nonlinear Schrödinger equations 21-dimensional NLS: iq t q xy ± 2q x 1 y q 2 =, 1 which have sister equations: iq t q xy ± x [2iq x 1 y q 2 ] =, 2 are such generalizations of the NLS in 21-dimensions. It is easy to see that equations 1 are the reductions of the 21-dimensional coupled nonlinear Schrödinger equation 21-dimensional CNLS see, for example, Ref. [11]: iq t q xy 2q x 1 y qr =, ir t r xy 2r x 1 y qr = 3 by taking r = q r = q respectively equations 2 are the reductions of the following 21-dimensional coupled derivative nonlinear Schrödinger equation 21- dimensional CDNLS: iq t q xy 2i x [q x 1 y qr] =, ir t r xy 2i x [r x 1 y qr] = 4 by taking r = q r = q respectively. Equations 1 belong to the class of equations discussed by Zakharov [12] re-derived by Strachan for κ = 1. [13] Some dynamical properties soliton solutions to Eqs. 1 or 2 were obtained in Ref. [11]. The gauge equivalent structure of the NLS both in dimensions were displayed in Refs. [14] [17]. Although a deep geometric understing of a differential equation describing p.s.s. or a differential system describing surfaces of constant nonzero Gaussian curvature has been displayed in Refs. [1], [3], [5] [7], [9], [17] [19], we know little about this for a 21-dimensional evolution equation or system. In this paper, we try to give a geometric characterization of 21-dimensional evolution integrable equations to see how the geometric approach may provides dynamical information for The project partially supported by National Natural Science Foundation of China

2 136 WU Jun-Yi, DING Qing, Keti Tenenblat Vol. 45 such equations. In fact, we show that there is a class of 21-dimensional integrable equations, such as the 21-dimensional CNLS 3 the 21-dimensional CDNLS 4, etc., describing 3-dimensional hyperbolic spaces 3-h.s.. As a direct consequence, the geometric construction of an infinite number of conservation laws of equations describing 3-h.s. is illustrated. Furthermore we display a new infinite number of conservation laws of the 21-dimensional nonlinear Schrödinger equation the derivative nonlinear Schrödinger equation by a geometric way. This paper is organized as follows. In Sec. 2, we introduce the notion of a 21-dimensional evolution model either equation or system describing 3-h.s. Then we show that the 21-dimensional CNLS, the 21- dimensional CDNLS, the 21-dimensional GHF model see Example 3 below describe 3-h.s. In Sec. 3, we construct an infinite number of conservation laws for evolution equations describing 3-h.s. by a geometric way, which is also illustrated by concrete examples. 2 Differential Systems Describing 3-Hyperbolic Spaces It is well known see, for examples, Refs. [3] [9] that a differential equation for a real-valued function ux, t, or a differential system for a vector-valued function ux, t, is said to describe pseudospherical surfaces p.s.s. resp. spherical surfaces s.s. if it is the necessary sufficient condition for the existence of smooth real functions f ij, 1 i 3, 1 j 2, depending only on u its derivatives, such that the 1-forms ω j = f j1 dx f j2 dt, ω 3 : = ω 12, 1 j 3, 5 satisfy the structure equations of a surface of constant i.e. dω Ω Ω =. curvature 1 resp. 1: dω 1 = ω 3 ω 2, dω 2 = ω 1 ω 3, dω 3 = δω 1 ω 2, 6 in which δ = 1 resp. δ = 1. Analogously, a 21- dimensional evolution equation for a real-vector-valued function ux, y, t is now said to describe 3-dimensional hyperbolic spaces 3-h.s. if it is the necessary sufficient condition for the existence of smooth real functions f j jk k, 1 j, k 3 fl, 1 j, k, l 3, depending only on u, its derivatives possibly its integro-differentials, such that the 1-forms ω j = f j 1 dx f j 2 dy f j 3 dt, ω jk = f jk 1 jk jk dx f2 dy f3 dt, ω jk ω kj =, 1 j, k 3 7 satisfy the structure equations of a 3-dimensional Riemannian manifold of constant sectional curvature 1, which is dω j = ω jk ω k, 1 j, k 3, dω jk = ω kl ω lj ω j ω k, 1 j, k, l 3. 8 Note that, if ω 1 ω 2 ω 3 on the domain W, on which u is defined, then ds 2 = ω 1 2 ω 2 2 ω 3 2 defines a Riemannian metric on W, whose Gaussian curvature is 1 the connection 1-forms are ω 12, ω 13, ω 23. This paper is based on the following key observation, which is a nontrivial generalization of the 11- dimensional case: Theorem 1 The structure equations 8 are equivalent to the integrability condition of the following Pfaffian system dψ = Ωψ, 9 in which ψ = Ω = 1 ω 2 iω 13 ω 1 ω 12 iω 32 ω 3, 1 2 ω 1 ω 12 iω 32 ω 3 ω 2 iω 13 Proof It is a straightforward computation. The sl2, C-1-form-valued matrix Ω may be regarded as a connection of a principal SL2, C bundle over R 3 ; the evolution equation expressed in this way shows the fact that the curvature dω Ω Ω of this connection vanishes. If we rewrite Eq. 9 as the form of Lax pair: ψ x = Ω 1 ψ, ψ y = Ω 2 ψ, ψ t = Ω 3 ψ, then Ω = Ω 1 dx Ω 2 dy Ω 3 dt, the integrability condition of Eq. 9 becomes Ω 1t Ω 3x [Ω 1, Ω 3 ] =, Ω 1y Ω 2x [Ω 1, Ω 2 ] =, Ω 2t Ω 3y [Ω 2, Ω 3 ] =. Apart from the above general considerations, we would like to give some concrete examples of such evolution equations. Example 1 The 21-dimensional CNLS. The 21-dimensional CNLS 3 permits the following Lax pair: F x x, y, t, η = iησ 3 UF x, y, t, η, F t x, y, t, η = 2ηF y x, y, t, η i{ x 1 y U 2 U y }σ 3 F x, y, t, η, in which η is a spectral parameter rx, y, t 1 U =, σ 3 =. qx, y, t 1 We let F y = V F, in which V = is a complex-valued matrix, i.e. v 1 = v1x, 1 y, t, η v1x,y,t,η u 1x,y,t,η u 2x,y,t,η v 1x,y,t,η iv1x, 2 y, t, η, u 1 = u 1 1x, y, t, η iu 2 1x, y, t, η, u 2 = u 1 2x, y, t, η iu 2 2x, y, t, η for some real functions v1, 1 v1, 2 u 1 1, u 2 1, u 1 2, u 2 2. The integrability conditions of F x = iησ 3 UF, F y = V F, F y = V F, F t = [2ηV i x 1 y U 2 ρ 2 U y σ 3 ]F lead respectively to the following equations for v 1, u 1, u 2 v 1x = qu 1 ru 2, u 1x = 2rv 1 2iηu 1 r y, u 2x = 2qv 1 2iηu 2 q y, 11 ψ1 ψ 2

3 No. 1 On Differential Equations Describing 3-Dimensional Hyperbolic Spaces 137 v 1t = i 1 x 2 yqr 2ηv 1y iq y u 1 ir y u 2, u 1t = ir yy 2ηu 1y 2ir y v 1 2iu 1 1 x y qr, u 2t = iq yy 2ηu 2y 2iq y v 1 2iu 2 x 1 y qr. 12 One may verify directly that the compatibility condition of Eqs is still exactly Eq. 3. This implies the existence of v 1, u 1, u 2. But, we cannot give explicit expressions of v 1, u 1, u 2. Now the corresponding sl2, C- 1-form-valued matrix Ω is Ω = iησ 3 Udx V dy [2ηV i 1 x CNLS describes 3-h.s., with ω j ω jk given as follows: ω 1 = q 1 r 1 dx u 1 1 u 1 2dy [2ηu 1 1 u 1 2 r 2 y q 2 y]dt, ω 2 = 2v 1 1 dy [4ηv x y q 1 r 2 q 2 r 1 ]dt, ω 3 = q 2 r 2 dx u 2 2 u 2 1dy [2ηu 2 2 u 2 1 q 1 y r 1 y]dt, ω 12 = q 1 r 1 dx u 1 2 u 1 1dy [2ηu 1 2 u 1 1 q 2 y r 2 y]dt, y U 2 U y σ 3 ]dt. Therefore, the 21-dimensional ω 13 = 2ηdx 2v 2 1 dy [4ηv x y q 1 r 1 q 2 r 2 ]dt, ω 23 = q 2 r 2 dx u 2 1 u 2 2dy [2ηu 2 2 u 2 1 r 1 y q 1 y]dt, in which q = q 1 iq 2 r = r 1 ir 2. Restricting the above discussion to the 21-dimensional NLS : iq t q xy 2q x 1 y q 2 = i.e. r = q, we see that the 21-dimensional NLS describes 3-h.s. with completely degenerated metric, i.e. ω 1 ω 2 ω 3 =. When y = x, the 21-dimensional NLS reduces to the NLS, meanwhile, the 1-forms ω j j = 1, 2, 3 vanish. The remaining 1-forms ω 12, ω 13, ω 23 satisfy the structure equations of a surface whose Gaussian curvature is 1. This coincides with the conclusion for the NLS displayed in Ref. [9]. Analogously, for the 21-dimensional NLS : iq t q xy 2q x 1 y q 2 = i.e. r = q, we see that the 21-dimensional NLS describes 3-h.s. still with completely degenerated metric. Similarly, when y = x, the corresponding 1-forms ω j j = 2, 12, 23 vanish the remaining 1-forms ω 1, ω 3, ω 13 satisfy the structure equations of p.s.s. This is just the statement for the NLS given in Ref. [9]. Example 2 The 21-dimensional CDNLS. The 21-dimensional CDNLS 4 permits the following Lax pair: F x = 1 iη 2 2ηr iη 2 x 1 y qr iηr y 2ηr 1 x y qr 2 2ηq iη 2 F, F t = ηf y iηq y 2ηq x 1 y qr iη 2 x 1 F. y qr Let F y = V F, in which V = is a complex-valued matrix. Therefore, 21-dimensional CDNLS v 1 1 iv2 1 u1 1 iu2 1 u 1 2 iu2 2 v1 1 iv2 1 describes 3-h.s., with the 1-forms ω j, ω jk given as follows: ω 1 = ηq 1 r 1 dx u 1 1 u 1 2dy {η 2 u 1 2 u 1 1 ηr 2 y q 2 y 2η[q 1 r 1 x 1 y q 1 r 1 q 2 r 2 q 2 r 2 x 1 y q 1 r 2 q 2 r 1 ]}dt, ω 2 = 2v1 1 dy [2η 2 v1 1 2η 2 x 1 y q 1 r 2 q 2 r 1 ]dt, ω 3 = ηq 2 r 2 dx u 2 2 u 2 1dy {η 2 u 2 2 u 2 1 ηq 1 y r 1 y 2η[q 1 r 1 x 1 y q 1 r 2 q 2 r 1 q 2 r 2 x 1 y q 1 r 1 q 2 r 2 ]}dt, ω 12 = ηq 1 r 1 dx u 1 2 u 1 1dy {η 2 u 1 2 u 1 1 ηq 2 y r 2 y 2η[q 1 r 1 x 1 y q 1 r 1 q 2 r 2 q 2 r 2 x 1 y q 1 r 2 q 2 r 1 ]}dt, ω 13 = η 2 dx 2v1 2 dy [2η 2 v1 2 2η 2 x 1 y q 1 r 1 q 2 r 2 ]dt, ω 23 = ηq 2 r 2 dx u 2 1 u 2 2dy {η 2 u 2 2 u 2 1 ηq 1 y r 1 y 2η[q 1 r 1 x 1 y q 1 r 2 q 2 r 1 q 2 r 2 x 1 y q 1 r 1 q 2 r 2 ]}dt, in which q = q 1 iq 2 r = r 1 ir 2. Example 3 The 21-dimensional GHF model. The following 21-dimensional HF model [2] 1 S t = 2i [S, S y] 2iuS, u x = 1 x 4i trss xs y, s3 s in which S = 1is 2 U2 s 1 is 2 s 3 U1 U1 with s2 1 s 2 2 s 2 3 = 1, is the reduction of the following model: 1 S t = 2i [S, S y] 2iσS, σ x = 1 x 4i trss xs y, 13 γ α GL in which S = β γ C 2 GL C 1 GL C 1 with γ2 αβ = 1, in which γ = γ α = β. Equation 13 will be called as the generalized 21-dimensional HF model 21-dimensional GHF model in this paper. The corresponding model in 11-dimensions was presented in Ref. [2]. The 21-dimensional GHF model permits a Lax pair as follows: F x = iηsf, F t = 2ηF y ηss y 2iσSF.

4 138 WU Jun-Yi, DING Qing, Keti Tenenblat Vol. 45 Thus the GHF model describes 3-h.s., with the 1-forms ω j, ω jk which can be explicitly written down as we did in Example 1 Example 2 by using the Lax pair above we omitted them here. 3 Conservation Laws Obtaining via Geometric Ways We may establish, as in Refs. [5] [18] for 11-dimensional p.s.s. resp. s.s. equations, local gauge equivalence between any two 21-dimensional equations describing 3-h.s. In an analogous geometric way by using Frobenius Theorem. However, it needs quite a lengthy computation. In this section, we shall concentrate ourselves on the geometric construction of an infinite number of conservation laws of the 21-dimensional equations describing 3-h.s. For a 11-dimensional integrable equation describing pseudo-spherical surfaces, one can obtain an infinite number of conservation laws by using the following geometric way due to Chern Tenenblat. [3,8] That is, given an arbitrary coframe {ω 1, ω 2 } a corresponding connection 1-form ω 12 on a smooth surface M equipped with the Riemannain metric ds 2 = ω 1 2 ω 2 2, there exists a new coframe {θ 1, θ 2 } a corresponding new connection 1-form θ 12 satisfying the structure equations: dθ 1 =, dθ 2 = θ 1 θ 12, θ 12 θ 2 =, 14 if the surface M is a pseudo-spherical surface. Then the old one-forms ω 1, ω 2, ω 12, the new one-forms θ 1, θ 2, θ 12 are connected by a rotation in an angle ρx, t: θ 1 = ω 1 cos ρ ω 2 sin ρ, θ 2 = ω 1 sin ρ ω 2 cos ρ, by a gauge transformation θ 12 = ω 12 dρ. It follows that θ 1, θ 2, θ 12 satisfy Eq. 6 with δ = 1 if only if the Pfaffian system ω 12 dρ ω 1 sin ρ ω 2 cos ρ = is completely integrable, one can check that this happens if M is a pseudo-spherical surface. This fact can be generalized to the present 3-dimensional case. In fact, we can get a new 1-form-valued matrix Ω with new 1-forms θ j, θ jk 1 j, k 3 obtained from Ω by an SL2, C gauge transformation as follows: Ω Ω = daa 1 AΩA 1 : = 1 θ 2 iθ 13 θ 1 θ 12 iθ 32 θ 3, 15 2 θ 1 θ 12 iθ 32 θ 3 θ 2 iθ 13 in which the general form of A is e iα/2 cosβ/2 sinβ/2 cosγ/2 i sinγ/2 A = e iα/2. sinβ/2 cosβ/2 i sinγ/2 cosγ/2 From Eq. 15, we obtain θ 1 = cos α cos βω 1 cos α sin β cos γ sin α sin γω 2 cos α sin β sin γ sin α cos γω 3, θ 2 = sin βω 1 cos β cos γω 2 cos β sin γω 3, θ 3 = sin α cos βω 1 sin α sin β cos γ cos α sin γω 2 sin α sin β sin γ cos α cos γω 3, θ 12 = cos α cos γ sin α sin β sin γω 12 cos α sin γ sin α sin β cos γω 13 sin α cos βω 23 sin α cos β dγ cos αdβ, θ 13 = cos β sin γω 12 cos β cos γω 13 sin βω 23 sin β dγ dα, θ 23 = sin α cos γ cos α sin β sin γω 12 sin α sin γ cos α sin β cos γω 13 cos α cos βω 23 cos α cos β dγ sin αdβ. For the purpose of simplicity, we take α = ρ, β =, γ =. Then, in this case, θ 1 = ω 1 cos ρ ω 3 sin ρ, θ 2 = ω 2, θ 3 = ω 1 sin ρ ω 3 cos ρ, θ 12 = ω 12 cos ρ ω 32 sin ρ, θ 13 = ω 13 dρ, θ 32 = ω 12 sin ρ ω 32 cos ρ. Because the given one-forms ω j, ω jk 1 j, k 3 satisfy Eq. 8, it is a straightforward computation to verify that dθ 12 θ 2 = ω 1 cos ρ ω 3 sin ρ θ 12 θ 2 ω 32 cos ρ ω 12 sin ρ θ 13 θ 3, dθ 13 θ 3 = ω 32 cos ρ ω 12 sin ρ θ 12 θ 2 ω 1 cos ρ ω 3 sin ρ θ 13 θ 3. Therefore the Pfaffian system θ 12 θ 2 =, θ 13 θ 3 =, i.e. dρ ω 13 ω 1 sin ρ ω 3 cos ρ =, ω 12 cos ρ ω 32 sin ρ ω 2 =, 16 is completely integrable by Frobenius theorem we may determine ρ from this integrable Pfaffian system 16. To sum up, we have Theorem 2 Given one-forms ω j, ω jk 1 j, k 3 satisfying the structure equations 8 of 3-dimensional hyperbolic spaces, there are new one-forms θ j, θ jk 1 j, k 3 obtained from ω j, ω jk 1 j, k 3 by a gauge matrix e iρ/2 A = e iρ/2 such that dθ 1 =, dθ 2 = θ 2 θ 1 θ 3 θ 1, θ 12 θ 2 =, dθ 3 = θ 3 θ 1 θ 32 θ 2, θ 13 θ 3 =, dθ 32 =, in which ρ is determined by integrable Pfaffian system 16.

5 No. 1 On Differential Equations Describing 3-Dimensional Hyperbolic Spaces 139 Now the dθ 1 dθ 32 above provide desired conservation laws of the equation under consideration. In fact, for a 21-dimensional equation describing 3-h.s., we see that the associated functions f j k, f jk l 1 j, k, l 3 can be formally exped as a power series of the spectral parameter η. The function ρx, y, t, η given by Eq. 16 can be also exped as a power series of η. The 1-forms θ 1 θ 32 then determine an infinite number of closed one-forms. One may note that this geometric approach is also suitable to equations describing 3-h.s. with the metric being degenerated, i.e. ω 1 ω 2 ω 3 =, as we shall see in the following examples. Example A The 21-dimensional NLS : iq t q xy 2q x 1 y q q = can be derived from 21-dimensional CNLS by taking r = q. It permits the following Lax pair: iη q i 1 x y q q i q y F x = F, F t = 2ηF y q iη iq y i x 1 F, 17 y q q in which η is a spectral parameter. We let F y = V F, in which ivx, y, t, η u 1 x, y, t, η iu 2 x, y, t, η V = u 1 x, y, t, η iu 2, x, y, t, η ivx, y, t, η in which v, u 1, u 2 are real functions satisfying the following equations: v x = 2q 2 u 1 2q 1 u 2, u 1 x = 2q 2 v 2ηu 2 qy 1, u 2 x = 2q 1 v 2ηu 1 qy 2, 18 v t = 2ηv y 2qyu 1 1 2qyu 2 2 x 1 yq q 2, u 1 t = 2ηu 1 y 2qyv 1 2u 2 x 1 y q q qyy 2, u 2 t = 2ηu 2 y 2qyv 2 2u 1 x 1 y q q qyy Now the corresponding sl2, C-1-form-valued matrix Ω, which is defined in Eq. 1, becomes Ω = iησ 3 Udx V dy [2ηV i x 1 y U 2 U y σ 3 ]dt. By a straightforward computation, we find ω 2 = ω 12 = ω 23 =, ω 1 = 2q 1 dx 2u 1 dy 4ηu 1 2qydt 2, ω 3 = 2q 2 dx 2u 2 dy 4ηu 2 2qydt 1, ω 13 = 2ηdx 2vdy [4ηv 2 x 1 y q q]dt. From Theorem 2, the following system is completely integrable the second equation of Eqs. 16 is automatically satisfied: ρ x = 2η 2q 1 sin ρ 2q 2 cos ρ, 2 ρ y = 2v 2u 1 sin ρ 2u 2 cos ρ, 21 ρ t = 4ηv 2 1 x y q q 4ηu 1 2q 2 y sin ρ 4ηu 2 2q 1 y cos ρ, 22 q 1 cos ρ q 2 sin ρ t = q 1 y sin ρ q 2 y cos ρ x 2ηq 1 cos ρ q 2 sin ρ y 23 provides conservation laws of 21-dimensional NLS. We suppose that v, u 1, u 2 can be formally exped as a power series of the spectral parameter η, so the function ρx, y, t, η given by Eqs can be also exped in power of η. The equation 23 then determine an infinite number of conservation laws of the 21-dimensional NLS. In the following we will derive the conservation laws compare them with other conservation laws derived before. In order to state the result, we need to fix our notation. We suppose v = v n x, y, tη n, u 1 = u 1 nx, y, tη n, u 2 = u 2 nx, y, tη n, 24 n= n= n= the functions ρ, sin ρ, cos ρ are of the form ρ = ρ n x, y, tη n, sin ρ = s n x, y, tη n, cos ρ = c n x, y, tη n, 25 in which n= ρ n = 1 n ρ η= n! η n, s n = 1 n sin ρ η= n! η n, c n = 1 n cos ρ η= n! η n. 26 From the definition we see that s = sin ρ, c = cos ρ, s n = 1 n sin ρ η= n! η n = 1 [ ρ n! η η ]η= cos ρ = 1 [ n 1! n! k!n 1 k! k1 ρ k1 k cos ρ ] η η= k = 1 η η= n k= c n = 1 n kρ k s n k, n 1. n k=1 n= n= n kρ k c n k, n 1, k=1

6 14 WU Jun-Yi, DING Qing, Keti Tenenblat Vol. 45 So the second system of Eqs. 26 can be rewritten as s = sin ρ, c = cos ρ, s n = 1 n kρ k c n k, n n 1, c n = 1 n k=1 n kρ k s n k, n Substituting Eq. 24 into Eqs , letting η =, then we obtain the equations for v, u 1, u 2 v x = 2q 2 u 1 2q 1 u 2, u 1 x = 2q2 v qy 1, u 2 x = 2q1 v qy 2, 28 v t = 2qyu 1 1 2qyu 2 2 x 1 yq q 2, u 1 t = 2qyv 1 2u 2 x 1 y q q qyy 2, u 2 t = 2q 2 yv 2u 1 1 x y q q q 1 yy. 29 From Eqs , we can derive the equations for ρ n n as follows: ρ x = 2q 1 sin ρ 2q 2 cos ρ, 3 ρ y = 2v 2u 1 sin ρ 2u 2 cos ρ, 31 ρ t = 2 1 x y q q 2q 1 y cos ρ 2q 2 y sin ρ, 32 ρ nx = 2δ n1 2q 1 s n 2q 2 c n, ρ ny = 2v n 2 k=1 n u 1 n ks k u 2 n kc k, k= ρ nt = 4v 4 u 1 ks k u 2 kc k 2qys 2 n 2qyc 1 n, k= in which n 1 δ ij is Kronecker delta. The equations for ρ n n 1 can be rewritten by substituting the expressions 27 as follows: ρ nx = 2q 1 cos ρ q 2 sin ρ ρ n 2 n 1 δ n1 [ jρ j q 1 c n j q 2 s n j ] 2δ n1, 33 ρ ny = 2u 1 cos ρ u 2 sin ρ ρ n 2 n 1 δ n1 [ jρ j u 1 c n j u 2 s n j ] 2v n 2 u 1 n ks k u 2 n kc k, 34 k= ρ nt = 2qy 1 sin ρ qy 2 cos ρ ρ n 2 n 1 δ n1 [ jρ j qys 1 n j qyc 2 n j ] 4v 4 u 1 ks k u 2 kc k. 35 k= One may verify directly that the compatibility conditions of Eqs are still exactly Eqs This implies the existence of the ρ. But we cannot give the explicit expression of ρ. And because equations 33, 34, 35 are integrable linear first-order partial differential equations for ρ n, we may get explicit expression of ρ n as follows: in which t ρ n = e a3x,y,t dt y e a2x,y,dy x e t e a3x,y,t dt y e t e a3x,y,t dt t const. e a2x,y,dy y a1x,,dx x b 3 x, y, t e t b 1 x,, e x a 1x,,dx dx b 2 x, y, e y a 2x,y,dy dy a3x,y,t dt dt t a3x,y,t dt y a2x,y,dy x a1x,,dx, a 1 = 2q 1 cos ρ 2q 2 sin ρ, a 2 = 2u 1 cos ρ 2u 2 sin ρ, a 3 = 2q 1 y sin ρ 2q 2 y cos ρ, b 1 = 2 n 1 δ n1 [ jρ j q 1 c n j q 2 s n j ] 2δ n1, b 2 = 2 n 1 δ n1 [ jρ j u 1 c n j u 2 s n j ] 2v n 2 u 1 n ks k u 2 n kc k, k=

7 No. 1 On Differential Equations Describing 3-Dimensional Hyperbolic Spaces 141 b 3 = 2 n 1 δ n1 [ jρ j qys 1 n j qyc 2 n j ] 4v 4 u 1 ks k u 2 kc k. Substituting Eqs into Eq. 23, then we have an infinite number of conservation laws, here we display only the first two of them q 1 cos ρ q 2 sin ρ t = q 1 y sin ρ q 2 y cos ρ x 36 [ ρ 1 q 1 sin ρ q 2 cos ρ ] t = [ ρ 1 q 1 y cos ρ q 2 y sin ρ ] x 2q 1 cos ρ q 2 sin ρ y, 37 in which ρ satisfies Eqs ρ 1 is given by ρ 1 = e 2 t q1 y sin ρ q2 y cos ρdt[ e 2 y u1 cos ρ u2 sin ρ t= dy e 2 x x y t 2 e 2 x q1 cos ρ q 2 sin ρ y=,t= dx dx e 2 y 2v 1 u 1 1 sin ρ u 2 1 cos ρ t= e 2 y 4v u 1 sin ρ u 2 cos ρ e 2 t const. e 2 y u1 cos ρ u2 sin ρ t= dy e 2 x k= q1 cos ρ q 2 sin ρ y=,t= dx u1 cos ρ u2 sin ρ t= dy u1 cos ρ u2 sin ρ t= dy dy q1 y sin ρ q2 y cos ρdt dt q1 cos ρ q 2 sin ρ y=,t= dx ]. Another sequence of conservation laws have been obtained in Ref. [11], which we will call the type I below, the first two of them are q q t = i[q q y x q x q y ], 38 q q x t = i[q y q x x q q xx q q 2 y ]. 39 It is easy to see that these two sequences of conservation laws are quite different from each other. In order to study the relation between these two sequences of conservation laws, we will explain briefly the method used in Ref. [11] to obtain the type I conservation laws below. Let ψ11 ψ 12 e iηx F = ψ 21 ψ 22 e iηx, from Eq. 17, we have ψ11x ψ 12x qψ 21 qψ 22 2iηψ 12 =. 4 ψ 21x ψ 22x qψ 11 2iηψ 21 qψ 12 Then, by taking ϕ = x ψ 11 /ψ 11 = n= ϕ n/ 2iη n1, transforming Eq. 17 into 2iηϕ = ϕ 2 q 2 q x, ϕ t = i y q ϕ q 2 ϕ 2ηϕ y i x q y, 41 q exping ϕ as series of 1/2iη, the authors got the type I conservation laws in Ref. [11]. Letting F y = V F, in which V = iv u 1 iu 2, we can verify directly that ϕ satisfies the following equation: u 1 iu 2 iv y = ū 2iv ϕ q ϕ q 2 ϕ q u. 42 Now set ψ = q e iρ, in which q is a solution to 21-dimensional NLS ρ solves Eqs Then, it is easy to verify that ψ satisfies Eqs too. We suppose ϕ = ψ 1/Φ, in which Φ is a function depending on x, y, t, η. Substituting this expression into Eqs , we see Φ x = 2iη 2ψ q x Φ 1, 43 q Φ y = 2iv 2uψ q y Φ ū q q, 44 Φ t = The general solution to Eqs is t 4iηv4ηuψ 2i Φ = e qȳ qxy q ψi q [ y e e y 4iηv 4ηuψ 2i q y dt 2iv2uψ qȳ q t= dy e 2iv2uψ qȳ q x t= dy y 2iη2ψ qx q u y q e q ψ i q xy q Φ y=,t= dx x 2η ū q i q y q e x 2iv2uψ qȳ q t= dy dy. 45 2iη2ψ qx q y=,t= dx dx

8 142 WU Jun-Yi, DING Qing, Keti Tenenblat Vol. 45 t 2η ū q i q t y q e const. e y 2iv2uψ qȳ q 4iηv4ηuψ 2i qȳ q t= dy e x qxy ψi q dt dt 2iη2ψ qx q y=,t= dx ]. Choosing a certain constant exping ψ 1/Φ as a power series of 1/2iη, we can obtain ϕ n for n 1 then the type I conservation laws. This indicates that the type I conservation laws can also be interpreted by geometric method. When y = x, the 21-dimensional NLS reduces to the NLS, V = iησ 3 U for F x = F y. So we have v = η, u 1 = q 1, u 2 = q 2. Then the first two conservation laws of the 21-dimensional NLS are reduced to q 1 cos ρ q 2 sin ρ t = qx 1 sin ρ qx 2 cos ρ x, [ ρ 1 q 1 sin ρ q 2 cos ρ ] t = [ρ 1 qx 1 2q 1 cos ρ ρ 1 qx 2 2q 2 sin ρ ] x, in which ρ satisfies ρ x = 2q 1 sin ρ 2q 2 cos ρ, ρ t = 2q q 2qx 1 cos ρ 2qx 2 sin ρ, ρ 1 = e 2 t q1 x sin ρ q2 x cos ρdt[ e 2 x x q1 cos ρ q 2 sin ρ t= dx 2 e 2 x q1 cos ρ q 2 sin ρ t= dx dx t 4 q 1 sin ρ q 2 cos ρ e 2 t const. e 2 x q1 cos ρ q 2 sin ρ t= dx ]. q1 x sin ρ q2 x cos ρdt dt These conservation laws are quite different from those found by Zakharov Shabat in Ref. [21], for those conservation laws in Ref. [21] can be deduced just by taking y = x from the type I conservation laws. Example B The 21-dimensional DNLS iq t q xy 2i x [q x 1 y q q] = can be derived from the 21- dimensional CDNLS by taking r = q. In a similar way as for NLS, we may obtain an infinite number of conservation laws, here we display only the first two of them: q 1 cos ρ q 2 sin ρ t = [qy 1 sin ρ qy 2 cos ρ 2q 1 cos ρ q 2 sin ρ x 1 y q q] x, [ ρ 1 q 1 sin ρ q 2 cos ρ ] t = [ ρ 1 q 1 y cos ρ q 2 y sin ρ 2ρ 1 q 1 sin ρ q 2 cos ρ 1 x y q q] x, in which ρ satisfies the following equations: ρ x =, ρ 1 = e 2 y u1 cos ρ u2 sin ρ t= dy[ x y t ρ t =, ρ y 2v 2u 1 sin ρ 2u 2 cos ρ =, 2q 1 sin ρ q 2 cos ρ y=,t= dx 2 v 1 u 1 1 sin ρ u 2 1 cos ρ t= e 2 y ] u1 cos ρ u2 sin ρ t= dy dy const. 2[ q 1 y cos ρ q 2 y sin ρ 2q 1 sin ρ q 2 cos ρ 1 x y q q]dt. in which the definitions of v i u k i i =, 1 k = 1, 2 are similar to those of v i u k i in Example A. Remark The geometric characterization of 21-dimensional integrable equations displayed in this paper provides a new mathematical way to investigate these equations. We believe that it will be useful in exploring further dynamical geometric properties of 21-dimensional integrable equations. References [1] R. Sasaki, Nucl. Phys. B [2] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Stud. Appl. Math [3] S.S. Chern K. Tenenblat, Stud. Appl. Math [4] C.H. Gu H.S. Hu, Sci. in China A [5] N. Kamran K. Tenenblat, J. Diff. Equa [6] E. Reyes, J. Diff. Equa [7] C. Rogers W.K. Schief, Stud. Appl. Math [8] J.A. Cavelcaute K. Tenenbalt, J. Math. Phys [9] Q. Ding K. Tenenbalt, J. Diff. Equa [1] L.D. Faddeev L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, Berlin, Heideberg [11] Z. Jiang R.K. Bullough, Phys. Lett. A [12] V.E. Zakharov, in Solitons, eds. P.K. Bullough P.J. Caudrey, Springer, Berlin 198. [13] I.A.B. Strachan, J. Math. Phys [14] Q. Ding, Phys. Lett. A [15] V.E. Zakharov L.A. Takhtajan, Theor. Math. Phys [16] Q. Ding, J. Phys. A: Math. Gen [17] M.L. Rabelo, Stud. Appl. Math [18] Q. Ding Z. Zhu, Sci. in China A [19] R. Myrzakulov, S. Vijayalakshmi, R.N. Syzdykova, M. Lakshmanan, J. Math. Phys [2] J. Honerkamp, J. Math. Phys [21] V.E. Zakharov A.B. Shabat, Soviet Phys. J.E.T.P