A Note on a Three-Dimensional Bäcklund Transformation

Transcription

1 Interntionl Journl of Applied Science nd Engineering 006., : 5-0 A ote on Three-Dimensionl Bäcklund Trnsformtion Xuncheng Hung * nd Edwrd H. C. Chng Yngzhou Poltechnic Universit-903 Ho Yue Yun, Moon PrkYngzhou, Jingsu 50 Chin tionl Tipei College of Business3 Section, Chi-n Rod, Tipei, Tiwn, R.O.C. Abstrct: The Bäcklund trnsformtion (BT) for three-dimensionl nonliner wve eqution nd its nonliner superposition formul re studied in this note. We prove tht the three dimensionl Bäcklund trnsformtion obtined b Leibbrndt, et l. cn be decomposed into three two-dimensionl BTs. Some results on the -dimensionl Liouville eqution re lso discussed in the rticle. Kewords: Bäcklund trnsformtion; Liouville eqution; superposition formul. Introduction For the pst decdes the stud of nonliner wve equtions hs ttrcted lot of ttentions from scientists nd mthemticins. Some powerful tools such s the singulr perturbtion, the inverse scttering trnsform, the Bäcklund trnsformtion (BT), etc., re developed in solving these equtions. It is interesting to note tht the Bäcklund trnsformtion ws first introduced in pseudo-sphericl surfce but now is ver useful in nonliner equtions. Liouville eqution in three dimensions tkes the form: = ep, = + +, () z with the following conditions: d dr, 0, r ( + + z ) + () In two-dimensionl spce, this eqution is reduced to ( ) χ ep( χ) + = k (3) with some boundr condition on χ, χ is sclr field, nd, k re rel constnts. Eqution (3) ws first obtined b Liouville, nd ws studied lter b mn well-known mthemticins tht include Picrd, Poincre nd Bierberbch, etc. The eqution hs significnt pplictions in electro-sttistics, hdrodnmics, cosmolog, isotherml gs spheres nd monopole theor. A Bäcklund trnsformtion for eqution () nd its nonliner superposition formul ws proposed b Leibbrndt, et l. [3,]. In this note, we prove tht the three-dimensionl Bäcklund trnsformtion of [3,] cn be decomposed into three two-dimensionl BTs. We lso discuss some results for the -dimensionl Liouville eqution. Since the theor of Bäcklund trnsformtions is still ver ctive [6,8], thus this discussion hs, ob- * Corresponding uthor; e-mil: h3@hoo.com Accepted for Publiction: September 9, Chong Universit of Technolog, ISS Int. J. Appl. Sci. Eng., 006., 5

2 Xuncheng Hung nd Edwrd H. C. Chng viousl, some interests.. Decomposition of Bäcklund trnsformtion for the Liouville eqution The Bäcklund trnsformtion derived b Leibbrnt et l. for the Liouville eqution in three sptil dimensions is + iβ Ki ( β ) = ep epiθ 3 z () K I + i( σ + σ ) (5) σ ep( iλσ ) (6), 0 0 i 0 σ =, σ =, σ3 = 0 i 0 0 re Puli mtrices nd I the identit mtri, θ, λ (0 θ π, 0 λ π ) re the prmeters of the BT, nd β re rel functions stisfing Eqution () nd the Lplce Eqution: = = + + (7) β 0, z respectivel. ote tht ep( iλσ ) = Icosλ+ iσ σ sin λ isin λ icos λ isin λ icos λ ep θ = I cosθ + sinθ icos λ sin λ icos λ isin λ (8) (9) We cn prove tht the Bäcklund trnsformtion of () tkes the following form: Ki ( β ) = ( A+ ib) ep (0) A= Icosθ () sinθ sin λ sinθcosλ B = sinθ cosλ sinθsin λ () This mtri eqution implies the four Equtions: ( + iz)( iβ ) = ( cosθ + isinθsin λ) ep (3) ( iz)( iβ ) = ( cosθ isinθsin λ) ep (3b) i ( iβ ) = isinθ cosλ ep (3c) i ( iβ ) = isinθ cosλ ep or, their equivlent forms: ( iβ ) = cosθep (3d) () ( iβ ) = sinθcosλep (b) z ( iβ ) = sinθsinλep (c) If we compre the rel nd imginr prts of the equtions in (), nd use the integrl conditions, we cn show tht equtions (,b) re Bäcklund trnsformtion for the two- dimensionl Liouville eqution in the light-cone coordinte sstem: = sinθcosθcos λep (5) nd the wve eqution β = 0. Similrl, equtions (,c) re BT for z = sinθcosθsin λep (6) nd the wve eqution β z = 0 ; nd equtions (b,c) re BT for z sin θsin λcos λep = (7) 6 Int. J. Appl. Sci. Eng., 006.,

3 A ote on Three-Dimensionl Bäcklund Trnsformtion nd β z = 0 We would like to point out tht: ). The equtions (5), (6), nd (7) cn be written, with some coordinte trnsformtions, s the stndrd form s in (3). ). From () we hve cos θ ep = (8) sin θcos λep = (8b) zz = sin θsin λep (8c) nd thus + + = ep (9) zz which is the originl three-dimensionl Liouville eqution. Similrl, we hve β + β + βzz = 0 (0) which is Lplce eqution (7). Therefore, we cn see tht the three sets of BTs for two-dimensionl Liouville equtions lso stisf the conditions for the three-dimensionl Liouville eqution (). In Bäcklund trnsformtion (), since the prmeters θ, λ (0 θ π, 0 λ π ) re rel, ll the trigonometric functions in () re rel. Therefore, we hve shown tht the Bäcklund trnsformtion () for the three-dimensionl Liouville eqution () is ble to be decomposed into three sets of BTs for two-dimensionl Liouville equtions. Similr results cn be obtined for the Liouville eqution in sptil dimensions. 3. Some results of Liouville equtions in sptil dimensions We write () into the following: () ( iβ ) = ep () ( iβ ) = ep () (b) (3) ( iβ ) = 3 ep (c) = ( θλ, ) ( =,,3) re the prmeters of the trnsformtion, which stisf + + = () 3 It is es to see tht, =,,3 re rel due to the rnges of θ, λ. Consider the trnsformtion in -dimensionl spce + iβ () ( iβ ) = ep (,,... ) =, =,,3,...,, stisf =. = Obviousl, nd β in (3) stisf the -dimensionl Liouville eqution ( ) = ep () = nd the -dimensionl Lplce eqution ( ) β = 0 (5) = Let ε = (,... ), nd etend it to stndrd orthogonl bsis of the -dimensionl spce R : { ε, ε... ε }. Consider the mtri A with the low vectors of ε ( =... ), nd the coordinte trnsformtion: () () ( ) T () () ( )... = A... (6) We hve ( ) ep, () iβ = (7) ( iβ ) = 0, ( =,3,..., ) (8) ( ) This mens tht under the new coordintes, nd β re onl relted to (), which stisf the regulr esil solvble two-dimensionl T Int. J. Appl. Sci. Eng., 006., 7

4 Xuncheng Hung nd Edwrd H. C. Chng Liouville nd wve equtions: = ep, (9) () () β = (30) 0. () () For the Liouville eqution with time vrible t, we hve similr result. For emple, consider ( t ) = ep, = + +z, (3) nd its Bäcklund trnsformtion Ki ( β ) = ep epiθ (3) K I + i( σ + σ ) + σ, (33) 3 z t [ i ] nd β stisfies the equtions σ ep ( λσ )ep( τσ ), ( < τ <+ ) (3) ( t ) β 0, = = + + (35) z It cn be proved tht the Bäcklund trnsformtion (3) with time vrible t hs similr structure with the one of () without time t.. onliner superposition formul of solutions In references [3,], the following nonliner superposition formul of solutions for Liouville eqution of higher dimensions hve been used: () () β β0 tn = R tnh, (36) θ, λ () θ, λ, (), 0, θ λ θ λ 0 iβ iβ iβ iβ (38) We will show tht the bove superposition formul is trivil. According to (), we cn chnge (38) into iβ iβ iβ iβ (39) () () 0, 0, θ, λ, It follows tht, to the rnge of θ nd ( =,,3) depend on re rel due λ. We hve () ( ) () 0 ( iβ0 ) = ep () ( ) () 0 ( iβ0 ) = ep () ( ) () iβ + ( iβ) = ep () ( ) () iβ + ( iβ) = ep, ( =,,3) From (0,c) nd (0b,d), we obtin = ( ) ( iβ0 iβ) () () 0 iβ + ep + ep nd = ( ) ( iβ0 iβ) () () 0 iβ + ep + ep Also, (0) (0b) (0c) (0d) () () ) R =± ( +ℵ ) /( ℵ, ℵ <, ℵ cosθ cosθ + sinθ sinθ cos( λ λ ), (), (37) (), β 0 nd β re the rel solutions of equtions () nd (7), respectivel. These solutions re connecting b the Bäcklund trnsformtion (). Assume tht the BT () is permutble, we hve () () () () 0 iβ + 0 sinh cosh () () sinh () () () () 0 iβ + 0 = sinh cosh () () sinh (3) 8 Int. J. Appl. Sci. Eng., 006.,

5 A ote on Three-Dimensionl Bäcklund Trnsformtion or equivlentl, () () () () 0 iβ + 0 sinh ep () () () () 0 iβ + 0 = sinh ep. () Eliminting the nonzero fctor () () + 0 ep, nd epnding the hperbolic sine functions ields tht () () iβ0 iβ ( + )sinh cosh () () iβ 0 iβ = ( )cosh sinh ; tht is, () () + β0 β tnh tnh i = Since tht tnh( iχ) = itn χ, we hve () () + β0 β tnh i tnh = (5) (6) (7) ow we check the formul (7) crefull. () () Since,, β0, β nd, re ll rel, the right hnd side of (7) is pure imginr while the left hnd side is rel. Thus, both vnish, or β = β 0. In other words, it is impossible to obtin new solution from the solution iβ0,, () () b the formul (7). Therefore, the formuls (37) nd (38) (the Bcklund trnsformtion in the references [3,]) re trivil, b which no new solutions will be produced. Topics relting to the Bäcklund trnsformtion nd nonliner superposition formuls for nonliner wve Equtions re ver ctive. And, in literture, there re some further discussion bsed on the formuls (37) nd (38) (see [5-8,,], for instnce). Therefore, the discussion in this note is, of course, necessr. References [ ] Ockendon, J. R. (Ed.) 003. Applied Prtil Differentil Equtions - Oford Tets in Applied nd Engineering Mthemtics, Oford Universit Press, Oford. [ ] Gesztes, F., Holden H., Bollobs B., Fulton W., Ktok A., Kirwn F., nd Srnk P Soliton Equtions nd their Algebro-Geometric Solutions: Vol., (+)-Dimensionl Continuous Models Cmbridge Studies in Advnced Mthemtics, Cmbridge Universit Press. [ 3] Leibbrndt, G., Wng S. S., nd Zmni. 98. Bäcklund generted solutions of Liouville s Eqution nd their grphicl representtions in three sptil dimensions, Journl of Mthemticl Phsics, 3, 9: [ ] Leibbrndt, G onliner superposition for Liouville s eqution in three sptil dimensions, Letters in Mthemticl Phsics, : [ 5] Hung, X. C Another method of deriving uto-bäcklund trnsformtions for nonliner evolution equtions, Journl of Phsics A-Mthemticl nd Generl, 6: [ 6] Hung, X. C A one-prmeter Bäcklund trnsformtion for the Eucliden Liouville nd wve equtions, Phsic Script, 7: 3-3. [ 7] Tu, M. H. 00. On the Bäcklund trnsformtion for the Mol Korteweg-de Vries hierrch, Journl of Phsics A-Mthemticl nd Generl, 3: [ 8] Bi, C. L. 00. Bäcklund trnsformtion nd multiple soliton solutions for the (3+)-dimensionl izhnik ovikov Veselov eqution, Chinese Phsics., 3: -. [ 9] Hu, X. B. nd Springel, J. 00. A Bäcklund trnsformtion nd nonliner superposition formul for the Int. J. Appl. Sci. Eng., 006., 9

6 Xuncheng Hung nd Edwrd H. C. Chng Lotk-Volterr hierrch, AZLAM Journl. : -8. [0] Rourke, D. E. 00. Elementr Bäcklund trnsformtions for discrete Ablowitz Ldik eigenvlue problem, Journl of Phsics A-Mthemticl nd Generl, 37,7: [] Tu, G. Z. nd Hung, X. C From ewton s lw to generlized Hmiltonin sstem (I) Some results on liner skew-smmetric opertor, Journl of Yngzhou Poltechnic Universit, 9, :36-6. [] Hung, X. C. nd Tu, G. Z A new hierrch of integrble sstem of +-dimensions From ewton s lw to generlized Hmiltonin sstem (II), Interntionl Journl of Mthemtics nd Mthemticl Sciences. (to be published) 0 Int. J. Appl. Sci. Eng., 006.,