Intrinsic Geometry of the NLS Equation and Heat System in 3-Dimensional Minkowski Space

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Intrinsic Geometry of the NLS Equation and Heat System in 3-Dimensional Minkowski Space"

θάνατος Καραβίας
6 χρόνια πριν
Προβολές:

1 Adv. Sudies Theor. Phys., Vol. 4, 2010, o. 11, Irisic Geomery of he NLS Equaio ad Hea Sysem i 3-Dimesioal Mikowski Space Nevi Gürüz Osmagazi Uiversiy, Mahemaics Deparme Eskişehir, Turkey guruz@ogu.edu.r Asrac The oliear Schrodiger of repulsive ype for imelike curves ad oliear hea sysem are derived i a geeral irisic geomeric seig icludig a ormal cogruece i 3-dimesioal Mikowski space. Mahemaics Sujec Classificaio: 53B99 Keywords: Mikowski space, NLS equaio 1 INTRODUCTION Defiiio 1.1 Le R 3 1 e a 3-dimesioal Mikowski space. If (x 1,x 2,x 3 ) ad (y 1,y 2,y 3 ) are he compoes of X ad Y wih respec o a allowale coordiae sysem, he X, Y = x 1 y 1 +x 2 y 2 x 3 y 3 which is called a Mikowski ier produc [1]. Theorem 1.1 Le γ(s) a ui speed curve i R 3 1, s eig he arclegh parameer.cosider he Free frame { = γ,,} aached o he curve γ = γ(s) such ha is is he ui age vecor field, is he pricipal ormal vecor field ad = ε 2 is he iormal vecor field. The Free -Serre formulas is give y 0 ε 2 κ 0 ε 1 κ 0 ε 3 τ 0 ε 2 τ 0 (1) where, = ε 1,, = ε 2,, = ε 3. is he semi Riemaia coecio o M ad κ = κ(s) ad τ = τ(s) are he curvaure ad he orsio fucios of γ, respecively.the iormal moio of imelike curves i he Mikowski 3-space is equilave o he oliear Schrödiger equaio (NLS ) of repulsive ype [2]

2 558 N. Gürüz iq + q 1 2 q, q 2 q = 0 (2) s where q = κ exp(i τd s). The iormal moio of spacelike curves i he Mikowski 0 3-space is equivale o he oliear hea sysem r = r ss + r 2 q, q = q ss q 2 r (3) s q = κ exp( τd s), r = κ exp( τd s) (4) s Irisic geomeric seig Irisic geomery he oliear Schrodiger equaio is sudied i E 3 y Rogers ad Schief [3]. A characerizaio hree dimesioal vecor field i respec of aholoomic coordiaes was iroduced y Vraceau [4], ad he followed Marris ad Passma [5] i E 3. Le Θ e a hree dimesioal vecor field i respec of aholoomic coordiaes i R 3 1. The,, is he age, pricipal ormal, ad iormal direcios o he vecor lies of Θ. Irisic derivaives of his orhoormal riad are give y followig s 0 ε 2 κ 0 ε 1 κ 0 ε 3 τ 0 ε 2 τ 0 (5) 0 ε 2 θ s ε 3 (ε 1 Γ τ) ε 2 θ s 0 ε 2 ε 3 div ε 1 (ε 1 Γ τ) div 0 (6), ad s 0 ε 2 (ε 1 Γ τ) ε 3 θ s ε 1 (ε 1 Γ τ) 0 (ε 1 κ + div) ε 1 θ s ε 2 ε 3 (ε 1 κ + div) 0 (7) show direcioal derivaives i he ageial, pricipal ormal ad iormal direcios i R 3 1. The (5) deoes he usual Free-Serre relaios, (6) ad (7) also deoe he direcioal derivaives of he orhoormal riad {,, } i he -ad -direcios i R 3 1. Therefore,

3 Irisic geomery of NLS equaio 559 grad = ε 1 s + ε 2 + ε 3. (8) ad θ s ad θ s are he quaiies which firs iroduced y Bjorgum i E 3 [6]. θ s =,, θ s =, (9) ad div = ε 1 s + ε 2 + ε 3, = ε 2 θ s + ε 3 θ s, (10) div = div = I addiio, ε 1 s + ε 2 + ε 3, = ε 1, ε 1 κ ε 3 τ + ε 2, = ε 1 κ + ε 3, ε 1 s + ε 2 = ε 1, s + ε 3 + ε 2,, + ε 3, + ε 3, = ε 2, (11) (12) where curl = ε 1 s + ε 2 + ε 3, = ε 1 (ε 2 κ)+ε 2 + ε 3 = ε 1 κ + ε 2 [ε 2, + ε 3, +ε 3 [ε 2, + ε 3, ] ]=ε 1 κ + ε 1 Ω s (13) Γ s = curl, = ε 1 ε 2 ε 3 (,, ) (14)

4 560 N. Gürüz is called he aormaliy of -field i R 3 1. curl = ε 1 s + ε 2 + ε 3, = ε 1 ( ε 1 κ ε 3 τ)+ε 2 + ε 3 = ε 1 ε 3 τ + ε 2 [ε 1, + ε 3 = ε 1 (τ +, = ε 1 ε 3 div + ε 2 Γ + ε 1 θ s, ] ε 1 ε 3, ) + ε 1 θ s + ε 1 ε 3 div (15) where Γ = curl, = ε 1 (τ +, ) (16) is he aormaliy of he -field. where curl = ε 1 s + ε 2 + ε 3, = ε 1 ε 2 τ + ε 2 [ε 1, + ε 2 +ε 3 [ε 1, + ε 2, ] = ε 1 (τ, ) ε 1 θ s + ε 1 ε 2 ε 3, = ε 2 (κ + ε 1 div) + ε 3 Γ ε 1 θ s Γ = curl, = ε 1 (τ, ] (17), ) (18) is he aormaliy of he -field. The ideiy cur lg radγ = 0 gives γ curl + grad(γ s s ) + γ cur l +grad(γ ) (19) + γ γ curl + grad( ) =0 s Wih usig he equaio (13),(15),(17) i (19), we have

5 Irisic geomery of NLS equaio 561 (ε 1 ε 3 2 γ ε 1ε 2 2 γ ) ( 2 γ ε s 1ε 2 γ 3 ) s (ε 1 ε 2 γ 2 2 γ ) s s = γ s Γ s ε 1 ε 3 γ div + ε 2 γ (κ + ε 1div) (20) = γ Γ + ε 1 γ θ s = ε 1κ γ s ε 1 γ θ s γ Γ Here s,, deoe aholoomic coordiaes. Fially oe has ie compaiiliy relaios oaied from he ideiies grad grads =0,grad grad =0, ad grad grad =0. Bu i is eough givig hree relaios for his work. κ + ε (ε 1 Γ τ) 1ε 3 s ) = θ s Γ θ s Γ + ε 1 τθ s (21) θ s s = ε 3 θ 2 s ε 1ε 3 (ε 1 Γ τ)γ + ε 2 ε 3 κ(ε 1 κ + div) ε 1 ε 2 ε 3 τ(ε 1 Γ τ) (22) (ε 1 κ+div) s + ε 1 τ = θ s ((ε 1 + ε 3 )κ + div) ε 1 ε 2 Γ div (23) 3 The Case Γ =0i R 3 1 The geomeric cosrai Γ = 0 (24) imposed. Here, our aim is o derive he oliear Schrodiger equaio of repulsive ype for imelike curves ad oliear hea sysem for spacelike curves y mea of such a cosrai. The cosrai (24) deoes he ecessary ad sufficie resricio for he exisece of a ormal cogruece of Π surfaces coaiig he s-lies ad -lies. If he s-lies ad -lies are ake as parameric curves o he memer surfaces U =cosa of he ormal cogruece, he he surface meric is give y Ad I U = ds 2 + g(s, )d 2. (25) grad U = ε 1 s + ε 3 = ε 1 + g 1/2 Hece, from (5) ad (7), we have (26) 0 ε 2 κ 0 ε 1 κ 0 ε 3 τ 0 ε 2 τ 0 (27)

6 562 N. Gürüz g 1/2 0 ε 2 τ ε 3 θ s ε 1 τ 0 ε 3 (ε 1 κ + div) ε 1 θ s ε 2 ε 3 (ε 1 κ + div) 0 Furhermore, if r shows he posiio vecor o he surface he. (28) r =, r = g1/2, (29) ad equaios (27) ad (28) implies ha ad Thus we oai 2 r = g1/2 (ε 2 τ+ ε 3 θ s ) (30) 2 r = ε 2g 1/2 τ+ g1/2. (31) l g 1/2 θ s = ε 3 (32) I he case Γ =0, he compaiiliy codiios Equaios (21)-(22) ecome he oliear sysem κ ε τ 1ε 3 = ε s 1 τθ s (33) θ s = ε s 3 θ 2 s + ε 2ε 3 κ(ε 1 κ + div)+ε 1 ε 2 ε 3 τ 2 (34) (ε τ s 1κ + div)+ε 1 = θ s ((ε 1 + ε 3 )κ + div)). (35) The Gauss- Maiardi-Codazzi equaios ecome wih (32) ε 1 τ + s (ε 1κ + div) = ε 3 ((ε 1 + ε 3 )κ + div)) s (g1/2 ) (36) (g 1/2 ) ss = ε 2 g 1/2 (κ(ε 1 κ + div)+ε 1 ε 3 τ 2 ) (37) 1 g 1/2 κ ε 1ε 3 τ = ε 1 ε 3 τ l g1/2 (38)

7 Irisic geomery of NLS equaio 563 (A) For he imelike curves: I his case ε 1 = 1,ε 2 =1,ε 3 =1. Equaios (36),(37) ad (38) ecome s (g1/2 ( κ + div)) + κ g1/2 s τ =0, (39) κ (gτ) g1/2 =0, (40) g 1/2 ss = g 1/2 (κ( κ + div)+τ 2 ). (41) Wih elimiaio of κ + div ewee (40) ad (41), we have τ = g 1/2 (τ2 ss )+κ κ (g1/2 ) (42) If we ow wrie g 1/2 = μκ where μ chages oly i he direcio ormal o he cogruece, he μ, hus Equaios (40) ad (42) reduces o κ =2κ s τ + κτ s (43) τ =(τ 2 κ ss κ + κ2 2 ) s (44) Wih usig equaios (43) (44), we have iq + q ss 1 2 q, q 2 q R()q =0, (45) where R() =(τ 2 κss + κ2 ) κ 2 s=s 0.This is oliear Schrodiger equaio of repulsive ype. (B) For he spacelike curves We cosider a spacelike curve wih imelike iormal. I his case ε 1 =1,ε 2 = 1,ε 3 = 1. Usig (43) ad (44), (4)we have { r = r ss + r 2 q q = q ss q 2 r. (46)

8 564 N. Gürüz This sysem is o liear hea sysem. REFERENCES [1] O Neill B, Semi Riemaia Geomery, [2] Dig Q, A oe o he NLS ad he Schrödiger flow of maps, Phys Le. A 248 (1998), [3] Rogers C, Schief W., Irisic geomery of he NLS equaio ad is auo Backlud rasformaio, Sudies i applied mahemaics 101: [4] M.G.Vraceau, Les espaces o-holoomes e leurs applicaios mecaiques, Mem. Sci.Mah.Z6: [5] A.W.Parris ad S.L.Passma, Vecor fields ad flows o developale surfaces, Arch.Ra. Mech Aal.32:29-86 (1969). [6] O.Bjorgum, O Belrami vecor fields ad flows, Par I. Uiversie I.Berge, Arok,1951, Naurvieskapelig rekke -1. [7] Hasimoo H., A soluio o a vorex filame, J Fluid Mech,51: , Received: March, 2010

Σχετικά έγγραφα

The Estimates of the Upper Bounds of Hausdorff Dimensions for the Global Attractor for a Class of Nonlinear

The Estimates of the Upper Bounds of Hausdorff Dimensions for the Global Attractor for a Class of Nonlinear Advaces i Pure Mahemaics 8 8 - hp://wwwscirporg/oural/apm ISSN Olie: 6-384 ISSN Pri: 6-368 The Esimaes of he Upper Bouds of Hausdorff Dimesios for he Global Aracor for a Class of Noliear Coupled Kirchhoff-Type