 Applied Mathematics, 2011, 2, 666-675 doi:10.4236/am.2011.26088 Published Online June 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Computation of Some Geometric Properties for New Nonlinear PDE Models Nassar Hassan Abdel-All1, Mohammed Abdullah Abdullah Hamad1, Mohammed Abdullah Abdel-Razek1, Amal Aboelwafa Khalil2 1Department of Mat hem at i cs, Faculty of Science, Assiut University, Assiut, Egypt 2Department of Mat hem at i cs, Faculty of Science, Sohag University, Sohag, Egypt E-mail: amalaboelwafay@yahoo.com Received March 16, 2011; revised March 25, 2011; accepted March 28, 2011 Abstract The purpose of the present work is to construct new geometrical models for motion of plane curve by Dar-boux transformations. We get nonlinear partial differential equations (PDE). We have obtained the exact so-lutions of the resulting equations using symmetry groups method. Also, the Gaussian and mean curvatures of Monge form of the soliton surfaces have been calculated and discussed. Keywords: Motion of Curve, Darboux Transformations, Gaussian and Mean Curvatures, Symmetry Groups 1. Introduction Kinematics of moving curves in two dimension is for- mulated in terms of intrinsic geometries. The velocity is assumed to be local in the sense that it is a functional of the curvature and its derivatives. Plane curves have received a great deal of attention from mathematics, physics, biology, dynamic system, image processing and computer vision [1,2]. Evolution of plane curve can be understand as a deformation of a plane curve in arbitrary direction according to arbitrary a mount in a conti- nuously process that has the time as a parameter. Phy- sically this arbitrary a mount is a function of velocity, so this process create a sequence of evolving planer curve moving by a funcion of velocity, the type of the motion (evolution) of this family of planer curves classified depend on our choice of velocity function. Let is the position vector of a curve C moving in space and let  denote respectively the unit tangent, principal normal and binormal vectors vary along . If we introduce the darboux vector (see ), ,rstC,,Tnb=Tkb (1) then the Serret-Frenet equations may be written as the following : ==,====sssTTknnnbkbbn  where s is the arc length parameter along , the curvature and Ck the torsion. In the present moving curve context, the time t enters into the system (2) as a sparameter. The general temporal evolution in which the triad ,,bTn remains orthonormal adopts the form (darboux formula)  =,==tttTnbnTbTbn (3) where  is the geodesic curvature,  is the normal curvature and  is the geodesic torsion. Here, it is required that the arc length and time derivatives com- mute. This implies inextensibility of . Accordingly, the compatibility conditions C=sttsTT , =stnnts and =stbtsb, applied to the systems (2) and (3) yield =,==.stsstkkk, (4) If the velocity vector =tr of a moving curve has the form C=Tnb, (5) then imposition of the condition yields stts rr ==sssss sttnn bb nb.   (6) ,T (2) Substitute about Serret-Frenet equations N. H. ABDEL-ALL ET AL.667 =0,=,=.ssskk  (7) The temporal evolution of the curvature and the torsion k of the curve may now be expressed in terms of the components of velocity C, and  by substitution of (7) into (4) to obtain =,=,tssts skkks    (8) where 1=.ssskk   (9) Motion in a plane occurs if =0 and =0. Then Equation (8) becomes ==tsssskk kk.ss (10) From Equation (7), we have then Equation (10) becomes =dks2=tss skkk d.ks (11) If we take then  d=d =ksFkFk dFkkd=d= dsks Fk= Fkks k, hence the equa- tion (11) becomes  2=.sstssskkkFkkFkkFkkk (12) 2. Symmetry Group Now, we want to present the most general Lie group of point transformations, which apply on obtaining equa- tions Definition 1. We consider a scalar order PDE represented by m thΔ,=0, where is natural numbermsk m (13) where =iss, is a vector for which the components i=1,,ips are independent variables and =jkk, is a vector cosest of =1,j,qjk dependent vari- ables, and =mmmkks. The infinitesimal generator of the one-parameter Lie group of transformations for equa- tion (13) is =1=,piiilXs ks1=q,sk,k (14) where ,isk, ,sk are the infinitesimals, and the th prolongation of the infinitesimal generator (14) is (see [5-8]) m=mlpr XX1=jjq,msk ,jk (15) where jl,=mskjD1=iipiki.,1=ijipk (16) and is the total derivative operator defined by Dkkkks=, =, =1,,jjkkjsDiijjjjp (17) A vector field X is an infinitesimal symmetry of the system of differential Equations (13) if and only if it satisfies the infinitesimal invariance condition =0=0mlpr X (18) 3. Soliton Geometry In this paper, we construct the soliton surfaces associated with the single soliton solutions (similarity solution) of the Equation (12). For this purpose, if =,kkst is a similarity solution of Equation (2), we have a solution surface  given from the Monge patch ,kst=,fs,t. The tangent vectors sftf, for the soliton surface  are given by =1,0, ,=0,1, .sttsfkfk (19) The normal unit vector field on the tangents pT is given by =.fsfNtstf (20) fThe and fundamental forms on 1st2nd are de- fined respectively by 221112 2222111222=d,d=d2ddd ,=d,d = d2ddd,Iff gsgstgtIIfNLsLstLt (21) where the tensor and are given by ijgijL11 122211 1222=,, =,, =,,=,, =,, =,.ss stttss stttgffgff gffLfNLfNLfN (22) The Gauss equations associated with  are 121111 11121212 12122222 22=,=,=,sss tsts ttts tfffLNfffLNfffLN (23) Copyright © 2011 SciRes. AM N. H. ABDEL-ALL ET AL. 668 while the Weingarten equations comprise 121222 1112 111112122222 12121211 22==,,sstsgLgLgL gLNfgggLgLgL gLNfggttff (24) where 2211 2212==st .gff ggg (25) The functions ij in (23) are the usual Christoffel symbols given by the relations ,,,1=2iiljjll jjlgg g g (26) The compatibility conditions ts=ss stff and ts =st ttff applied to the linear Gauss system (23) produce the nonlinear Mainardi-Codazzi system 2221112 1112222212111112212 1112222212112=2=tsstLLL LLggg ggLLL LLggg gg0,0,(27) or, equivalently, 121111211 121212112211121122211 221222122212=,=,tstsLL LLLLL LLL22 (28) The Gaussian and mean curvatures at the regular points on the soliton surface are given by respectively 211 221212 211 2212===, LL LLKkk gggg g0 (29) 11 221212221112 211 2212211==22LgLg LgHkk ggg. (30) where =det ijgg, =det ijLLk and 12 are the principal curvatures. The surface for which is called parabalic surface, but if 1 and 2 constant or 1constant and 2, we have surface semi round semi flat (cylinderial like surface).The integrability conditions for the systems (2) and (3) are equivalente to the Mainardi-Codazzi system of PDE (27). This give a geometric interpretation for the surface defined by the variables , kkK=0=0kk=0==kts, to be a soliton surface [9,10]. 4. Applications 4.1. Case I: =Fkk The Equation (12) becomes 32341=322=ts ssssssskkkkk kkkkk0. (31) The infinitesimal point symmetry of Equations (31) will be a vector field of the general form =Xstk (32) on 3=MR; our task is to determine which particular coefficient functions , and  are functions of the variables ,st and and will produce infinitesimal symmetries. In order to apply condition (18), we must compute the third order prolongation of which is the vector field k,X3pr= JJ,JXstk k    (33) whose coefficients, in view of (31), are given by the explicit formulae =,=,=,=.ssstssstttststttss sss tssssstsssssss tssssssstDkkkkDkkkkDkkkkDkkkk      (34) The vector field X is an infinitesimal symmetry of the Equation (31) if and only if 33212234pr= 0=333 32 682=0.tssss ssss ssss sssssssXkkkkkkkkk kkkkkk    (35) Substituting the prolongation Formulae (34), and equa- ting the coefficients of the independent derivative mono- mials to zero, leads to the infinitesimal determining equations which together with their differential conse- quences reduce to the system 11=, =, ====022tstk k stk. (36) The general solution of this system is readily found 31 32311=, =, =22cs cctcck, (37) where the coefficients ic are arbitrary constants. Therefore, Equation (31) admits the three-dimensional Lie algebra of infinitesimal symmetries, spanned by the three vector fields 12311=, =, =22XXXstk.ststk  (38) The combination of space and time translations 12XX=ysct lead to a reduction of (31) to an ordinary differential equation (ODE) by the transformation  and =kwy where c is the speed of the travelling wave. That is Copyright © 2011 SciRes. AM N. H. ABDEL-ALL ET AL. Copyright © 2011 SciRes. AM 6694=0.4.2. Case II: 1=Fkk 233322wwwwwcwwwww  (39) Now, the solution of the Equation (39) is, 32211d2ln2 2wycwwcwcwc=0, (40) In this case Equation (12), becomes 52 3912=tssss ssskk kkkkkk 0, (45) where 12 and 3 are the integration constants, if we consider it equal zero, hence the solution of Equation (39) becomes ,cc cLie point symmetry for this equation is given by 1234=, =,1=,=3XXstXs kXtk,skt22 2222==11ccwcy csct. (41) k (46) this solution is a similar solution to Equation (31), This solution is in the Monge form ==,wwy wst which define a regular surface as show in Figure 1 (). =1, 15, 0.12cs tThe combination 12==XXcX cst gives rise to travelling wave solutions a wave speed . The vector field cX has invariants and which reduces (45) to the ODE =ysct=kwThis surface is a soliton surface . From (29) and (30), one can see that the Gaussian and mean curvatures of the soliton surface ( ) are given by respectively 1125912wwwww cwww 3=0. (47) 321241241363=0, =,32ttssKH ts  2 (42) Now, solving the equation with the Lie symmetry spanned by 12=,2=.3YyYy wyw (48) where  22187246862252324424 2462246=12=18366441 7 83783107 660708933 499157.ttssttssssst stsstss stsstsssstsss   (43) If we take the vector field we obtain solution 2Y23=wy, and substituting in Equation (47) we get w132=9c and the solution 132329=cksct (49) The symmetry generator 311=22Xst kstk  leads to invariants 2=tys and . These the in- =wskvariants transform Equation (31) to the following ODE, 3222 323 23242383624243616 421=0''ywwywyww wywwywwyw ywwww   (44) Remark 1. For regularity the parameters of the soliton surface must be satisfied sct, i.e., for =sct we have singularit y cuspidedge as shown in Figure 3. The Gaussian and mean curvatures respectively are (shown in Equation (50)) If we take the vector field 13XX we here the in- riants and =yt1=1kThe numerical solution of Equation (44) is shown in Figure 2 (intial condition and ).  1=1,1=2ww1=3wws, that is then =0w=wconstant and   733102 2111132 233333333=0,270 6= .818 68 68124324381KstHst ttstststsstsst  (50) N. H. ABDEL-ALL ET AL. 670 Figure 1. Soliton surfaces of (41). =1ak,s (51) thus we have Figure 4. The vector field leads to the invariants 4Xsy= and the tarnsformation 13=ktw reduces (45) to an ODE in the form 6227336= 0,wwwwwww 3 (52) this equation can be solved numericaly (intial condition and  0=2, 0=2ww0=3w ) as shown in Fig- ure 5. 4.3. Case III: =21Fk k In this case Equation (12) takes the form 62 3422440 =tssss sssskkkkkkkk kk 0, (53) Lie point symmetry of this equation is given by 123=,=,=XX Xstk,stst  k (54) The combination 12==XcX Xcst gives rise to travelling wave solutions with wave speed c. The vector field X has invariants and which reduces (53) to =ysct=wk62 34224 40=cwwwwww wwww solving Equation (55) hence 3242216d241824 9wycwcwcwcw=0, (56) If we take the integration constants to be zero hence the solution takes the form 2222==88wcy csct , (57) For regularity the parameters of the soliton surface must be satisfied 18sct c at 1, i.e., for =cc1=8sct c we have singularity (cuspidaledge) as shown in Figure 6. 0. (55) Figure 2. Numerical solution of (44). Copyright © 2011 SciRes. AM N. H. ABDEL-ALL ET AL.671 Figure 3. Soliton surfaces of (49). Figure 4. Solution of (51). Gaussian and mean curvatures are 2222222148 242=0, =8ttss tstsKHst    (58) where  162242422422=8=818896246 641889615 4415sttts s tsststtst tstssst s  Figure 5. Numerical solution of (52). Figure 6. Soliton surface of (57). The symmetry generator 3=Xstkstk leads to the invariants and . After some detailed and tedious calculations, (53) becomes ODE =yst=wsk3222 362 232 432182472 364014= 0''ywwywyww wwywwywyw yww w  (60) 522 (59) Copyright © 2011 SciRes. AM N. H. ABDEL-ALL ET AL. 672 The numerical solution of Equation (60) is shown in Figure 7 (intial condition and ).  1=1,1=2ww1=3''w 4.4. Case IV: =11Fk k In this case Equation (12) becomes 76543433323332 254642821229123=0,tttt tsss4ssss sssss ssssss sssss skkkkkkkk kkkkkkkkkkk kkkkkkk kkkkkkk kkk  (61) Lie point symmetry of this equation is given by 12=, =XXst (62) The combination 12==XcX Xcst gives rise to travelling wave solutions with wave speed c. The vector field X has invariants and =ysct=wk Figure 7. Numerical solution of (60). which reduces (62) to 765434333233432 2546428212 29123cwwcwwcwwcwwcwwwwwwwwwww wwwwwww wwwwwwwwww=0       (63) solving the Equation (63) we get 31d1wycww =0 (64) where      2212 222212222221=342 ln2ln12ln 122ln24ln22 4ln4ln14ln 12ln2ln2 22ln12ln1 2,wcc wwcwwww cwwwcwww cwwcwwcw wwwwcwcc wcw c   (65) If we take , 1=cc 2=1c and 3=0c21d=1122wywwcw cw 0, (66) then 1212124=tan 2441164 .tan 24145wywwwww (67) Hence, we have a soliton surface given by the implicit equation 1212124=tan 2441164 ,tan 24145wsct wwwww (68) Gaussian and mean curvatures of implicit surface are  22322 34562=0,11141812=21 1211124Kcw ww wwHcw www w 3 (69) This surface is illustrated as in Figure 8. 4.5. Case V: =lnFkk In this case Equation (12) becomes Copyright © 2011 SciRes. AM N. H. ABDEL-ALL ET AL.673 Figure 8. Soliton surface of (68). 4234 466 lntssss ssssskk kkkkkkkkkkk =0 (70) Lie point symmetry for this equation is given by 123=,=, =XX Xtstk,stst  k (71) The combination 12==XcX Xcst gives rise to travelling wave solutions with wave speed c. The vector field X has invariants and wkich reduces (70) to =ysct=wk2424466lnww wwwcwwwww w =0. (72) We have solved the Equation (72) with intial condition ( and ) numerically, which represented in Figures 9(a), (b). The Figure 9(a) represents the numerical solution at the forword wave, while Figure 9(b) at the backword wave.  0=2, 0=2ww0=3w 4.6. Case VI: =ekFk  In this case Equation (12) becomes 42 23323433232 =0,ktss ssssssss sssekkkkskkkkk kkkk kkkkkk (73) Lie point symmetry for this equation is given by (a) (b) Figure 9. Numerical solution of (72). 12=, =XX,st (74) the travelling wave solution is obtained by Copyright © 2011 SciRes. AM N. H. ABDEL-ALL ET AL. 674 =Xcst by which (73) becomes the o.d.e (with the new independent variable , being the speed of the wave) =ysctc2222422 4 33322 =wwwww wwwwcewwwwww w 0,(75) by numerical (intial condition and , range of to 5) Figures 10(a), (b), This solution represents a curve on the soliton surface and .  0=2, 0=2ww0=3w=wws0=ct ysct 4.7. Case VII: =Fkk In this case Equation (12) becomes 723284181512= 0,tssss sssskk kkkkkkkk 4 (76) Lie point symmetry for this equation is given by 12322=,=, =33XXXstk,stst  k (77) the travelling wave solution is obtained by =Xcst by which (76) becomes the ODE (with the new independent variable , being the speed of the wave) =ysctc7222418 1512='wwwwwcwww w  40, (78) solving the Equation (78) we get 379432221d=22wwycwcwwcwcw0, (79) If we take the integration constants to be zero Equation (79) becomes 32292422=,32wcwcwycwwcw (80) we have a soliton surface given by the implicit equation 32292422=,32wcwcwsctcwwcw  (81) Gaussian and mean curvatures of implicit surface are 522377=0,174=,Kcc wH224221412 2cwcww (82)    (a) (b) Figure 10. Numerical solution of (75). Copyright © 2011 SciRes. AM N. H. ABDEL-ALL ET AL. Copyright © 2011 SciRes. AM 675 Figure 11. Soliton surface of (81). this surface is illustrated in Figure 11. 5. Conclusions We have discussed motion of curves in a plane and analysed nonlinear equations and related generalisations like vector using symmetry methods. These lead to exact solutions like travelling wave, soliton and other simi- larity solutions. Gaussian curvature equal zero and mean curvature don’t equal zero lead to be surfaces cylinder of  B. Kimia, A. Tannenbaum and S. Zucker, “On the Evolu-tion of Curves via a Function of Curvature. I. The Clas-sical Case,” Journal Mathematical Analysis and Applica-tion, Vol. 163, No. 2, 1992, pp. 438-458. doi:10.1016/0022-247X(92)90260-Kthese equations. 6. References  B. Kimia, A. Tannenbaum and S. Zucker, “Shapes, Shochs, and Deformations I: The Components of Two-Dimen-sional Shape and the Reaction-Diffusion Space,” Interna-tional Journal of Computer Vision, Vol. 15, No. 3, 1995, pp. 189-224. doi:10.1007/BF01451741  C. Rogers and W. K. 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