Motion of Space Curves in Three-dimensional Minkowski Space $R_1^{3}$, SO(2,1) Spin Equation and Defocusing Nonlinear Schr"odinger Equation

Motion of Space Curv es in Three-dimensional Mink o wski Space R 3 1 , SO( 2,1) Spin Equatio n and Defo cusing Nonlinear Sc hr¨ odinger Equation GOP AL MUNIRAJA Department of Mathematics, Bishop Cotton W omen’s Christian College, Bangalo r e-560 027, India. M LAKSHMANAN Cent er for Nonlinear Dynamics, Bha rathidasan Univ ersity ,Tiruchirapalli 620 024, India Abstract. W e consider the dynamics of moving curves in three-dimens io nal Minko wsk i space R 3 1 and deduce the ev olution equatio ns for the curv atur e and tor- sion of the curv e. Next by mapping a co ntin uous SO(2,1) Heisenber g spin chain on the space curve in R 3 1 , w e sho w that the defocusing nonlinear Schr¨ odinger equation(NLSE) can b e ident ified with the s pin chain, thereby g iving a geometrical interpretation of it. The a sso ciated linear eigenv alue problem is also obtained in a g eometrical w ay . Keyw ords : Mink owski space, F r enet equations, SO(2,1 ) Heisenberg spin equation, defo cusing nonlinear Sch r¨ odinger eq ua tion 2 1. INTRODUCTION Mo delling of ph ysical systems b y curv es, surfaces and other differen tial geometric ob j ects is highly rew arding, see for example the pioneering w ork of Hasimoto [1] on v ortex filamen ts and the one-dimensional con tinuum Heisen b erg ferromagnetic spin equation b y Lakshmanan, Ruijgork and Thompson [2,3 ]. In b oth cases the systems were sho wn to b e equiv alen t to the integrable nonlinear Schr¨ odinger equation (NLSE) of the fo cusing t yp e [1-5]. In recen t times the relation b etw een differen tial geometry and certain dynamical systems described by nonlinear ev olution equations in (1+1) and (2+1) dimensions, esp ecially the in tegr a ble systems , has come in to sharp fo cus [6- 13]. In tegra ble nonlinear ev olution equations o ccur in man y branc hes of ph ysics and applied mathematics. Such equations p ossess a num b er o f in teresting prop erties suc h as soliton solutions, infinite n umber of conserv a t io n laws, infinite num b er of symmetries, B¨ acklu nd and Darb oux transformations, bi-Hamiltonian structures and so on, see [7, 8]. No w it is we ll kno wn that a class of impo r t an t soliton equations can b e in terpreted in terms o f mo ving space curv es in R 3 and the linear eigen v alue problems of the soliton equations can b e obtained from the defining Serret-F renet equations of space curv es [6 , 9-13]. Extens ion to (2+1) dimensions is also p ossible. A brief surv ey of the dev elopmen ts up to more recen t times can b e seen in [13]. Our fo cus in this Letter is on the defo cusing NLSE, in whic h the sign of the nonlinear term is negative, whic h is encoun tered in man y phys ical problems. It was sho wn to b e in tegrable b y the in v erse scattering transform method[14] and admits dark soliton 3 solutions. How ev er, there do es no t see m to b e av ailable a simple differen tial geometric mo del and its equiv alen t spin sys tem for the defocusing case suc h as av a ilable fo r t he fo cusing NLSE as demonstrated in [1 ,2]. In this Letter w e wish to address this problem. W e ha ve found the ric h curv e theory a v ailable in three-dimensional Mink ows ki space is more suited to this problem. Nak a yama [16] has used the geometry of curv es on a three-dimensional ellipsoid in a four-dimensional Mink o wski space to obtain a mo del for the defo cusing NLSE. F or other related w orks see also [17 ,18]. But we feel our approac h is simpler and more direct and can b e extended profitably to o ther nonlinear ev olution equations a lso. In the next section w e giv e the basic equations of curv es in R 3 1 and fix the notation closely fo llowing [19] and [2 0]. Subsequen t sections establish the curv e mo del of the solution of the defo cusing NLSE with the to ols and tec hniques similar to the ones fo und in [3] and [20-2 3 ] and its connection to the SO (2 ,1) con tin uous Heisen b erg spin chain. W e hop e to extend similar treatmen t of in tegrable and nonin tegrable systems to surfaces in R 3 1 and higher dimensional Mink o wski geometries in o ur subsequen t w ork. 2. Motion of curv es in the Mink owski Space R 3 1 The nature o f the metric in a Mink owski space induces a ric h geometry of curv es a nd surfaces. F or instance the familiar Serret-F r enet equations in the Euclide an space R 3 giv e wa y to four suc h sy stems in R 3 1 . In this section w e giv e the basic curv e geometry apparatus in R 3 1 , see [19], and we closely follo w [20] in writing down the curv e ev olution 4 equations. The metric on the Mink ow ski space R 3 1 is giv en b y ds 2 = -dx 2 1 + dx 2 2 + dx 2 3 . W e note here that the scalar and vec tor pro ducts of t w o vec tors a = a 1 i + a 2 j + a 3 k , b = b 1 i + b 2 j + b 3 k in R 3 1 , where i, j, k are unit v ectors along the x, y , z axes resp ectiv ely , are give n as follo ws: Scalar Pro duct: a.b = g ( a, b ) = − a 1 b 1 + a 2 b 2 + a 3 b 3 . V ector Pro duct: a ∧ b =              − i j k a 1 a 2 a 3 b 1 b 2 b 3              , ( Note : In Mink o wski space a ve ctor a is defined to b e a unit v ector if g ( a, a ) = ± 1. A v ector a is said to b e sp ac e li ke if g ( a, a ) > 0, time like if g ( a, a ) < 0 and lig ht like or a nul l ve ctor if g ( a, a ) = 0 ). No w, let( e 1 , e 2 , e 3 ) b e the Serret-F renet frame of a unit speed (non-null) curv e α ( x ) in R 3 1 . Here e 1 is the unit ta ng en t v ector field , e 2 is the normal and e 3 is the binormal to α ( x ). L et g( e 1 , e 1 ) = ǫ 0 = ± 1, g( e 2 , e 2 ) = ǫ 1 = ± 1. Then g ( e 3 , e 3 ) = - ǫ 0 ǫ 1 . Then the Serret-F renet equations are given b y [19] e 1 x = ǫ 1 κ ( x ) e 2 , e 2 x = − ǫ 0 κ ( x ) e 1 − ǫ 0 ǫ 1 τ ( x ) e 3 , e 3 x = − ǫ 1 τ ( x ) e 2 . (1) Here τ and κ denote the torsion a nd curv ature resp ectiv ely of t he given space curv e α . 5 Using the vec tor pro duct relations e 1 ∧ e 2 = e 3 , e 2 ∧ e 3 = − ǫ 1 e 1 , e 3 ∧ e 1 = − ǫ 0 e 2 , (2) the Serret-F rene t equations (1) can b e compactly written as e ix = D ∧ e i , i = 1 , 2 , 3 , (3) where D is the Darb oux v ector defined as D = − ǫ 0 ǫ 1 τ e 1 − ǫ 0 ǫ 1 κe 3 . (4) No w, let us consider t he time ev olution of t he curve α ( x, t ). W e define a n angular momen tum like ve ctor Ω = Σ ω i e i , i = 1,2,3, whic h giv es the time ev olution o f the Serret-F renet system as e i t = Ω ∧ e i , i = 1 , 2 , 3 . (5) F rom (2) and (5) w e obtain e 1 t = − ǫ 0 ω 3 e 2 − ω 2 e 3 , e 2 t = ǫ 1 ω 3 e 1 + ω 1 e 3 , e 3 t = − ǫ 1 ω 2 e 1 + ǫ 0 ω 1 e 2 . (6) In order that the a b o v e tw o definitions are compatible w e require tha t ( e i ) x t = ( e i ) t x , i = 1 , 2 , 3 . (7) 6 F rom (1), (6 ) and (7) we obtain κ t = τ ω 2 − ǫ 0 ǫ 1 ω 3 x , τ t = ǫ 1 κω 2 − ǫ 0 ǫ 1 ω 1 x , ω 2 x = ǫ 1 τ ω 3 − ǫ 1 κω 1 . (8) The ab o v e equations constitute the ev olution of the curv ature and to r sion associated with an arbitrary curv e movin g in R 3 1 . 3. SO(2,1) Heisen b erg Spin E quation and Mapping to a Space Curve in R 3 1 Consider now the SO(2,1) Heisen b erg spin equation given b y S t = S × S xx , (9) where S is a unit vec tor in R 3 1 , t ha t is − S 2 1 + S 2 2 + S 2 3 = ± 1 . W e identify S with the unit tangent v ector e 1 of α ( x ). Then w e obtain from the spin equation e 1 t = e 1 × e 1 xx = e 1 × e (1 x ) x = e 1 × ( ǫ 1 κe 2 ) x = ǫ 1 e 1 × ( κ x e 2 + κe 2 x ) Hence from (1 ) and (2) and noting ǫ 2 0 = ǫ 2 1 = 1 w e hav e e 1 t = ǫ 1 κ x e 3 − κτ e 2 . (10) Next w e ha v e e 2 = ǫ 1 e 1 x κ from (1). Hence e 2 t = ǫ 1 e 1 xt κ − e 1 x κ 2 κ t . Using (1) and (10) we obtain e 2 t = [ ǫ 1 ǫ 0 κ 2 τ e 1 − ǫ 1 (2 κ x τ + κτ x + ǫ 1 κ t ) e 2 + ( κ xx + ǫ 0 κτ 2 ) e 3 ] /κ. (11) 7 Similarly we can deduce that e 3 t = κ x e 1 + [( ǫ 0 κ xx + κτ 2 ) e 2 − ǫ 1 (2 κ x τ + κτ x + ǫ 1 κ t ) e 3 ] /κ. (12) Comparing the ab ov e with (6) w e immediately obtain the ev olution equation for the curv ature as κ t = − ǫ 1 (2 κ x τ + κτ x ) . (13) The compatiblit y condition(7) applied to e 3 yields the ev olution equation for torsion as τ t = − ǫ 2 1 κκ x − ǫ 0 ǫ 1 ( κ xx κ + ǫ 0 τ 2 ) x . (14) The ab ov e equations define the ev olutio n of curv ature and torsion of the curv e associated with an SO (2,1) contin uum Heisen b erg spin system in R 3 1 . 4. Mapping on to the Defo cusing NLSE Let us first consider the case where ǫ 0 = − 1 and ǫ 1 = 1. Then equations (11) and (12) reduce to κ t = − 2 κ x τ − κτ x (15) and τ t = − κκ x + ( κ xx κ − τ 2 ) x . (16) W e now mak e the complex transformation u = κ 2 e i R x −∞ τ dx . (17) Then using (17) equations (15) and (1 6) are t r ansformed in to iu t + u xx − 2 | u | 2 u = 0 , (18) 8 whic h is nothing but the defo cusing nonlinear Sc hr¨ odinger equation. Now w e assume the energy a nd curren t densities of the spin system to b e related to the curv ature and torsion resp ectiv ely as ǫ ( x, t ) = 1 2 ∂ S ∂ x . ∂ S ∂ x = 1 2 κ 2 , (19) I ( x, t ) = S.S x ∧ S xx (20) so that the contin uit y given b y ǫ t − I x = 0 is satisfied. This contin uit y equation can b e easily sho wn to b e compatible with (15). Finally w e also observ e that the case of ǫ 0 = 1 and ǫ 1 = − 1 yields the solution of fo cusing NLSE under the condition that the curv e has a constan t to rsion. The other t w o cases of ǫ 0 = − 1 , ǫ 1 = − 1 and ǫ 0 = 1 , ǫ 1 = 1 do not reduce either t o the defo cusing or to the fo cusing NLSE for the tr a nsformation giv en by (17). 5. Reduction to AKNS Eigenv alue Problem Corresp onding to the Serret-F renet frame giv en by (1) (for ǫ 0 = − 1 and ǫ 1 = 1) w e define a new scalar v aria ble z l = e 2 l + ie 3 l 1 − ie 1 l , l=1,2 ,3, following [20, 21 ], from whic h w e obtain z lx = − iτ z l + iκ 2 (1 + z 2 l ) . (21) No w differen tiat ing z l with resp ect to t and usin g (5 ) , and after some detailed calculations, we arriv e at z lt = − iω 1 z l + ω 2 + iω 3 2 − ( ω 2 − iω 3 ) z 2 l 2 . (22) 9 Equations (21) a nd (22 ) are nothing but the Riccati equations. Ag ain the compatibilit y of (21) a nd (22) , that is ( z l ) xt = ( z l ) tx , leads to the corr ect eq uations fo r κ ( x, t ) and τ ( x, t ) as in ( 1 5) and (16). Defining z l = v 2 v 1 , equation (21) can b e written as v 1 x = iτ 2 v 1 − iκ 2 v 2 , v 2 x = iκ 2 v 1 − iτ 2 v 2 . (23) Similarly , from Eq.(22) w e obtain v 1 t = iω 1 2 v 1 − i 2 ( ω 3 + iω 2 ) v 2 , v 2 t = − iω 1 2 v 2 + i 2 ( ω 3 − iω 2 ) v 1 . (24) Using the compatibilit y conditions ( v ix ) t = ( v it ) x , i = 1 , 2 , from (23 ), (24) ab o v e w e once again get back easily the original equations for κ ( x, t ) and τ ( x, t ) . No w in tro ducing a suitable Galilean tra nsfor ma t io n and a gauge tra nsformation in to (2 3), (24) w e obtain the linear eigen v alue problem ψ 1 x = uψ 2 − iλψ 1 , ψ 2 x = u ∗ ψ 1 + iλψ 2 , (25) and the time ev olution of the eigenfunction as ψ 1 t = Aψ 1 + B ψ 2 , ψ 2 t = C ψ 1 + D ψ 2 , (26) 10 where A = − 2 iλ 2 − iuu ∗ , B = 2 uλ + iu x , C = 2 u ∗ λ − iu ∗ x . (27) and u is as defined in Eq. (17 ) . 6. Conclusions In this pap er, w e hav e sho wn how the dynamics of mo ving curv es in three dimensional Mink ows ki space R 3 1 can b e related to the dynamics of SO(2,1) spin equations and soliton equations of defo cusing NLS type. 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