Lf Scnc Journl 4;(5 hp://www.lfcnc.com Evoluon of Hlx Curv by obrvn vlocy Nr H. Abdl-All, H. S. Abdl-Azz, M. A. Abdl-Rzk, A. A. Khll 4, Dprmn of Mhmc, Fculy of Scnc, Au Unvry, Au, Eyp.,4 Dprmn of Mhmc, Fculy of Scnc, Soh Unvry, Soh, Eyp. E-ml: mlbolwf@yhoo.com (A. A. Khll ( Abrc: In h ppr, h quon of moon for nrl hlx curv r drvd by pplyn h fr compbly condon for dpndn vrbl ( m nd rc lnh. A pplcon of h quon of moon, mkdv quon olvd un ymmry mhod. [Nr H. Abdl-All, M. A. Abdl-Rzk, H. S. Abdl-Azz, A. A. Khll. Evoluon of Hlx Curv by obrvn vlocy. Lf Sc J 4;(5:4-47]. (ISSN:97-85. hp://www.lfcnc.com. 8 Kyword: Moon of curv, Gnrl hlx curv, Symmry of PDE, L lorhm.. Inroducon A lo of phycl proc cn b modlld n rm of h moon of curv, ncludn h dynmc of vorx flmn n flud dynmc [], h rowh of dndrc cryl n pln [], nd mor nrlly, h plnr moon of nrfc []. Th Subc of how pc curv volv n m of r nr nd h bn nvd by mny uhor. Ponrn work rbud o Hmoo who howd n [] h non-lnr Schrؤodnr quon dcrbn h moon of n old nonrchn hn vorx flmn. Lmb [4] ud h Hmoo rnformon o connc ohr moon of curv o h mkdv nd n-gordon quon. Nkym, l [5] obnd h n-gordon quon by condrn non-locl moon. Alo Nkym nd Wd [6] prnd nrl formulon of volvn curv n wo dmnon nd conncon o mkdv hrrchy. Nr, l [7]-[] hv udd voluon of mnfold nd obnd mny nrn rul. R. Mukhr nd R. Blkrhnn [] ppld hr mhod o h n- Gordon quon nd obnd lnk o fv nw cl of pc curv, n ddon o h wo whch wr found by Lmb [4]. For ch cl, hy dplyd h rch vry of movn curv ocd wh h on-olon, h brhr, h woolon nd h olon-nolon o-luon. In h ppr, Tm voluon quon for nrl hlx curv r drvd from pplyn h fr compbly condon for dpndn vrbl ( m nd rc lnh wll nrl hlx pc curv rconrucd from curvur. Hr, w condr h moon of curv n hr-dmnonl Eucldn pc. L r dno pon on pc curv. A uul, m dnod by. Th convnonl omrcl modl pcfd by vlocy fld, r ( Hr,, nd r h un nnl, norml nd bnorml vcor lon h curv, nd nd r h nnl, norml nd bnorml vloc. vloc fld r funconl of h nrnc qun of curv, for xmpl, curvur,, oron,, mrc,, c. Tm voluon quon for uch qun r drvd from ( nd h omrcl rlon. A pplcon o phyc, h modl r uful o dcrb h moon of vorx flmn n nvcd flud, moon of fron n vcou fnrn n Hl-Shw cll, nd knmc of nrfc n cryl rowh. Condr curv n -D rprnd by h prmr u.., r r( u. L r ( u, b h poon vcor of ny pon mov on h curv h m uch h r ( u, r( u. W dfn h mrc of h curv, ( u,, nd rc lnh ( u,, r r u :, ( u, ( v, dv, u u nd hn h un nn vcor of h curv, dfnd by r r :. u, Wh h dfnon of h un nn vcor on cn cnonclly dfn un norml vcor, nd bnorml vcor, ccordn o h wll-known Srr-Frn rlon 4
Lf Scnc Journl 4;(5 hp://www.lfcnc.com E AE ( whr E nd A, h curvur of h curv nd oron. In h m pr on cn lo condr dcrbn h m voluon of h curv. Fr w no h h FSE Eq. ( cn b wrn compcly ;,,. ( or Whr h Drboux vcor, mply (,,, n h, b. Th dynmc of h rd cn b dcrbd by dfnn nw vcor (,, uch h, mlr o Eq. (, ;,,. (4 whch cn b wrn n mrx form whr E BE B (5. Rconrucon of curv from curvur nd oron Condr h Srr-Frn rlon de AE (6 d whr ( E, A ( (, ( h curvur of h curv nd oron. Now, l (, (, (, (7 Bc xnc nd unqu rul for ym of lnr ODE urn h follown fundmnl horm: Thorm If ( nd ( r vn mooh funcon on n nrvl I (, b, whr I, nd ( > hn, vn, (6 h unqu oluon on I fyn (7. Morovr, r( r( ( d, (8 W hv nod h knowld of nd nlly fx pc curv nd w hr l om mpl funcon for nd nd h corrpondn curv hy nr. ( If ( (, h curv n ( unwd rh ln. ( If (, nd ( conn h curv crculr rc. ( If (, bu ( h curv wd rh ln. (v If ( conn nd ( conn, h curv crculr hlx. Th curv wnd round crculr cylndr. ( (v If conn, h curv nrlzd ( ( no ncrly crculr hlx. Th curv wnd round nrlzd crculr cylndr. W wll u powrful mhod clld nvlu mhod o olv h homonou ym ( (6n h c (v,.., w olv ( de ( BE (9 d wh B, Th d o fnd oluon of form ( E ( v ( ( d. ( Now kn drvv on E (, w hv de( ( v ( d ( Pu ( nd ( no h homonou quon (9, w 4
Lf Scnc Journl 4;(5 hp://www.lfcnc.com de( ( v ( d Bv ( So Bv v, B( E( ( whch ndc h mu b n nvlu of B nd v n oc nvcor. W fnd h,,, r h nvlu of B wh ocd nvcor E v (, E, v, v (, E ( r lnrly ndpndn ( vcor oluon of h homonou ym (9.Thn h nrl oluon E h ( of cn b wrn E ( [ c ( v h c ( v co( ( v co( ( v n( ( ] c whch cn b wrn n mrx form co( n( E h ( n( co( co( n( n( ( If (,,, (,,, ( v c c c ( nd (,, r h ndrd un vcor hn c, c, nd c Hnc r( r( co( d, n( d,. (. Equon of moon I mporn o noc h u nd r ndpndn bu nd r no ndpndn. A conqunc, wl u nd drvv commu, nd drvv n nrl do no commu; Lmm Proof. (4 u u Applyn h fr compbly condon ( (4 o h mrx E nd vcor r rpcvly, yld h follown quon: A B [ A, B] A (5 ( Wrn xplcly, Eq. (5 rd ;, (6 ( ( From h bov quon ( ( ( (7 Thorm If h dynmc of h curv r ( u,, vn by r Thn h moon of h curv dcrbd by 4
Lf Scnc Journl 4;(5 hp://www.lfcnc.com ( ( ( ( [ ( ( ( ( (8 For vn nd,,,, h moon of h curv drmnd from h quon. 4. Hlx C Hr w rrc ourlv o rc lnh prmrzd nrl hlx curv. Th. whch mpl h h moon of h curv dcrbd by If w ; ( ( nd, (9 hn from h compbly condon (4, (9ld o nd. whch mpl h h moon of h hlx curv dcrbd by, ( or (, ( whch ld o ( ( ] To rduc o h wo-dmnonl c,,, n whch mpl ( rduc o ( W cn dl wh h moon of hlx curv ( n dffrn wy from h n h prvou con. In rm of h componn of h nn, norml nd b-norml vcor, of hlx curv of h curv xprd ( ( co n n co,, co n whr d, Applyn h fr compbly condon ( (4 o h vcor r, yld h follown quon co co n n co n co n co n ( Th of quon nlly quvln o (. Th quon of moon for hrdmnonl curv rprnd by componn xplcly n (. To rduc o h wo-dmnonl c,,, n whch mpl ( rduc o co co co n n n n co whch quvln o (. (4 44
Lf Scnc Journl 4;(5 hp://www.lfcnc.com 5. Applcon mkdv quon In h bov quon (f w k w,, h known mkdv quon, (5 nd. whr Accordn o L lorhm [5]-[7], h nfnml nror of h mxml ymmry roup dmd by (5 vn by X (,, (,, (6 f nd only f h nvrnc condon of (5 [] X ( (7 whr X [] (5 X (8 h prolonon of h vcor fld (6. Th vrbl r vn by h formul :... D (, D D (, (9 D... D ( u,... Excun h L lorhm, w obn h L pon ymmr of (5 vn by X, X, X. ( W look for oluon nvrn undr h lnr combnon X X whr X X conn. Solvn h chrcrc ym for h nvrn of h lnr combnon, w obn, (, (, ( whr n rbrry funcon of. Th ubuon of ( no (5 yld ( ( ( ( ( ( ( whr prm dno dffrnon wh rpc o nd rprn h wv pd. Inron of ( onc ld o, k. ( Mulplyn boh d of ( by nd nrn onc mor w obn 4 ( k k. 6 (4 Fnlly, from (4 w hv k k d 4 6 d (5 whr k nd k r rbrry conn. Inrn boh d of h l quon w obn h nrl rvln wv oluon o (5 n mplc form d 4 k k 6 k k 5. Fr prculr c (6 Sn n (5w obn h follown KdV of hhr ordr, (7 If w um h,,, whn, n h nly prnd n h prvou con, hn (6 rduc o d 6 k for n rbrry conn k. Wh h ubuon (8 6 ch fnlly w obn h follown olon oluon o (8 6 (, ch ( k4, (9 whr k 4 conn of nron. 45
Lf Scnc Journl 4;(5 hp://www.lfcnc.com [ Curv corppondn o (4.5] Fur : Surfc corrpondn o (9, 6, [,], 5. Scond prculr c [,.] If w n h bov c hn h quon (7 rduc o, (4 nd (9 rduc o (, 6ch ( k4, (4 6, k4 hn (, 6ch( 6, If w (4 In h c w cn conruc h curv from curvur r( co( ( d, n( ( d (4 whr 6ch ( 6 d 6rcn ( nh( 6 [ Curv corppondn o (4 ] Fur : Curv of cond c wh [.8,.8] 5. Thrd prculr c If w,, hn h quon undr udy (5 rduc o (44, whch h nrlzd KDV quon. Procdn n h fr c,hn (6 rduc o d Un ubuon k ch (45 fnlly w obn h follown olon oluon o (45 (, ch ( k4, (46 4 [ Curv corppondn o (4 ] Fur : Surfc corrpondn o (46 46
Lf Scnc Journl 4;(5 hp://www.lfcnc.com 4,, [, ], Corrpondn Auhor: Dr. Aml Khll Dprmn of Mhmc Fculy of Scnc Soh Unvry- Eyp. E-ml: mlbolwf@yhoo.com [,.5] Rfrnc. H. Hmoo, J. Flud Mch. 5 (97 477.. R. C. Browr, D. A. Klr, J. Koplk nd H. Lvn,Phy. Rv. A 9 (984 5.. J. A. Shn, J. Dff. Gom. (99 4. G. L. Lmb, J. Mh. Phy. 8 (977 654. 5. Kzuk وNkym Hrvy Sur nd mk Wd Inrbly nd moon of curv, Phycl Rvw Lr Vol.69.No.8 (99, 7-. 6. Kzuk Nkym nd mk Wd Moon of curv n h pln, Journl of Th Phycl Socy of Jpn Vol.6.No. (99, 47-479. 7. Nr H. Abdl-All nd M. T. Al-dory, Moon of Curv Spcfd by Acclron n Fld n R, Appld Mhmcl Scnc, Vol. 7, (, no. 69, 4-48 8. Nr H. Abdl-All nd M. T. Al-dory, Moon of hypr urfc, Au unv. Journl of Mh. nd compur cnc 4 ( 9. 9. Nr H. Abdl-All, M. A. Abdl-Rzk, H. S. Abdl-Azz, A. A. Khll, Gomry of volvn pln curv problm v l roup nly, Sud n Mhmcl Scnc ( 5-6.. Nr H. Abdl-All, M. A. A. Hmd, M. A. Abdl-Rzk, A. A. Khll, Compuon of Som Gomrc Propr for Nw Nonlnr PDE Modl, Appld Mhmc ( 666-675.. M. A. Solmn, Nr. H. Abdl-All, Sod. A. Hn nd E. Dh, Snulr of Gu Mp of Pdl Hyprurfc n n. Lf Scnc Journl, (;8(4-7.. Nr H. Abdl-All, R. A. Hun nd Th Youf,Evoluon of Curv v Th Vloc of Th Movn Frm, J. Mh. Compu. Sc. ( 7-85.. R. Mukhr nd R. BlkrhnnPhy. L. A 7 (8 647. 4. R. Courn nd D. HlbrMhod of Mhmcl Phyc Volum,Nw York, N.Y.: Inrcnc Publhr, (96. 5. G. W. Blumn nd S. Kum.Symmr nd Dffrnl Equon, Sprnr, (989. 6. P. J. Olvr.Applcon of L Group o Dffrnl Equon, Sprnr, (986. 7. L. V. Ovnnkov.Group Anly of Dffrnl Equon, Acdmc, (98. /9/4 47