Degenerate Solutions of the Nonlinear Self-Dual Network Equation

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1 Commu Theor Phys 7 (09 8 Vol 7 o Jauary 09 Degeerate Solutios of the oliear Self-Dual etwork Equatio Yig-Yag Qiu ( 邱迎阳 Jig-Sog He ( 贺劲松 ad Mao-Hua Li ( 李茂华 Departmet of Mathematics igbo Uiversity igbo 35 Chia (Received Jue 08; revised mauscript received September 08 Abstract The -fold Darboux trasformatio (DT T [] of the oliear self-dual etwork equatio is give i terms of the determiat represetatio The elemets i determiats are composed of the eigevalues λ j (j = ad the correspodig eigefuctios of the associated Lax equatio Usig this represetatio the -solito solutios of the oliear self-dual etwork equatio are give from the zero seed solutio by the -fold DT A geeral form of the -degeerate solito is costructed from the determiats of -solito by a special limit λ j λ ad by usig the higher-order Taylor expasio For -degeerate ad 3-degeerate solitos approximate orbits are give aalytically which provide excellet fit of exact trajectories These orbits have a time-depedet phase shift amely l(t DOI: 0088/053-60/7// Key words: oliear self-dual etwork equatio Darboux trasformatio solito degeerate solutio Itroductio The studies of the discrete itegrable systems were iitiated i the middle of 970s Hirota had discretized various itegrable equatios such as the oliear partial differece KdV equatio [] the discrete-time Toda equatio [] the discrete Sie-Gordo (SG equatio [3] the Liouville s equatio [4] ad the Bäcklud trasformatio of the discrete-time Toda equatio [5] based o the biliear trasformatio methods Followig those works the discrete oliear systems may be used to the diverse areas to describe such physical situatios as the rogue waves i optical fibers ad water taks [6] ad the geeral rogue waves i the focusig ad defocusig Ablowitz- Ladik equatios [7 8] I the ladder type electric circuit coectio type electric circuit i coectio with the propagatio of electrical sigals is the followig oliear self-dual etwork equatio [9 0] I t = (+I (V V V t = (+V (I I + ( where I = I( t ad V = V ( t are voltage ad curret i the -th capacitace ad iductace respectively the fuctios of the discrete variable ad time variable t I t = di /dt ad V t = dv /dt Durig the recet decades fidig aalytical ad explicit solutios are oe of the most aspect i the studies of discrete systems [ 4] There are quite a few methods of the oliear itegrable systems such as the iverse scatterig trasformatio the biliear trasformatio methods of Hirota the dressig method the Bäcklud ad the Darboux trasformatio (DT the algebraic curve method [5 7] It is kow that the DT [7] is a powerful tool ot oly for the cotiuous itegrable system which geerates ew solutios of a itegrable equatio from a seed solutios but also is useful for the discrete itegrable system Matveev gave the discrete DT of the differetial-differece equatio [8] From the o the various DT of the discrete itegrable system are becomig more ad more importat for the discrete problems [9 0] I recet years the determiat represetatio T is widely used to get degeerate solitos ad breathers [ 3] by a limit λ j λ ad eve get rogue waves by a double degeeratio with the help of limit λ j λ λ 0 [4 6] Although there are some kow solutios [7] of Eq ( the determiat represetatio T [] of the -fold DT has ot bee give i literature This fact ispires us to costruct T [] of Eq ( The study of the degeerate solito ca go back to Zakharov who has show that the distace of two peaks i a -degeerate solito icreases with time like l(4η t as t [8] The positos [9 30] of KdV was firstly itroduced by Matveev which is a special degeerate solito ivolved a positive eigevalue The posito solutios have may iterestig properties that differet from the other of solito solutios The the posito solutios have bee costructed for may models such as the defocusig mkdv equatio [3 3] the SG equatio [33] ad the Todalattice [34] Moreover posito ca also be give by Hirota method ad the limit of eigevalues λ j λ [35] It is easy to see from Refs [8] [36] that the essetial feature of the above solutios is the degeeratio of eigevalues This fact motivates us to costruct the -degeerate solito of the oliear self-dual etwork equatio by degeerate limit of eigevalues λ j λ (j = 3 accordig Supported by the atural Sciece Foudatio of Zhejiag Provice uder Grat o LY5A00005 the atural Sciece Foudatio of igbo uder Grat o 08A6097 the SF of Chia uder Grat o 679 KC Wog Maga Fud i igbo Uiversity Correspodig author limaohua@bueduc c 09 Chiese Physical Society ad IOP Publishig Ltd

2 Commuicatios i Theoretical Physics Vol 7 to the T [] ad further discuss its properties This paper is orgaized as follows I Sec a determiat represetatio T [] of the -fold DT for the oliear self-dual etwork equatio is give The ew solutios of I [] ad V [] geerated by T [] are also provided explicitly by determiats I Sec 3 the -degeerate solitos I ds ad V ds are obtaied from -solitos by a limit λ j λ amely the degeeratio of eigevalues Our coclusios ad discussios are provided i the last sectio Determiat Represetatio of DT First the Lax pair of the oliear self-dual etwork equatio is as follows [7] Eψ = U ψ = λ + I V ψ t = W ψ = λ I V I + V λ I λv λ + I V I + V λ I λv λ I V ψ ( ψ (3 where ψ =(ψ ψ T is the vector eigefuctio λ is the eigevalue parameter idepedet of ad t the shift operator E is defied by Ef( t = f( + t = f + Z t R From the compatibility coditio of the oliear self-dual etwork equatio ψ t = ψ t we ca obtai the zero curvature equatio U t = (EW U U W (4 It is easy to fid that Eq (4 is compatible with the Eq ( by the calculatios ext the discrete DT for Eq ( is give o the basis of Eqs d (3 So we itroduce a -fold gauge trasformatio such that ψ [] ψ [] = T [] ψ (5 + = U [] ψ [] ψ [] t = W [] ψ [] (6 where U [] ad W [] have the same forms as U ad W respectively Further we set T [] to preserve the compatibility coditio of the Lax equatio amely U [] = T + U T W [] = (T t + T W T (7 Thus T [] is also called a -fold Darboux trasformatio of Eq ( The mai task i this sectio is to get the determiat represetatio of T [] i terms of the eigefuctios associated with the seed solutio Let = set oe-fold DT as T [] = b ( a 0 λ + b 0 c d c 0 d 0 (8 ( 0 T [] (λ = b 0 a 0 λ + a b c d a 0 b 0 c 0 ad d 0 are ukow fuctios of ad t Ad the we are goig to substitute it ito Eq (7 By comparig the coefficiets of λ i (i = 0 it is trivial to get a 0 = d b 0 = c At the same time we set d 0 = a = i order to simplify the later calculatios The the T [] becomes ( ( 0 a T [] 0 = λ + b 0 (9 0 b 0 a 0 The a 0 ad b 0 ca be give from its kerel T [] (λ (ψ = 0 (0 eigefuctio ψ is i the form of [ ] ψ ψ (λ = ψ Solvig Eq (0 yields λ ψ ψ ψ λ ψ a 0 ψ = λ ψ b 0 λ = ψ ψ W W where the matrix ( ψ ψ W W (ψ = λ ψ λ ψ Geerally all elemets i T [] give by Eq (9 are - order determiats I order to get uified ad coveiet determiat represetatio we fid that all elemets of ca be expressed by followig 3-order determiats i umerators T [] T [] = T [] (λ; λ 0 λ where = W W ˆξ ˆξ 0 λ 0 W ( λ ψ ˆξ = ψ 0 0 W ˆξ λ 0 ( W ˆξ I this case we are goig to make sure that each elemet of the matrix is a determiat cosistig of the same submatrix W The represetatio give i Eq ( is more helpful to uderstad clearly ad accurately the process for iteratio of T [] i order to get the -fold DT It is clear that ψ [] = T [] (λ ψ = 0 Similarly the -fold DT T [] is give by iteratig T [] twice which should be the followig form: b b a λ + 0 b 0 0 Solvig the algebraic equatios give by the its kerel ie T [] (λ j ψ j = 0(j = the T [] (λ = T [] (λ; λ λ = 0 λ 0 λ 0 0 λ 0 W ˆξ W ˆξ W 0 λ 0 λ 0 λ 0 λ 0 W ˆξ W ˆξ

3 o Commuicatios i Theoretical Physics 3 ψ ψ λ ψ λ ψ λ W W (ψ ψ = ψ λ ψ λ ψ λ ψ ψ ψ λ ψ λ ψ ˆξ = λ ψ λ ψ λ ψ λ ψ It is easily to check that ψ [] j = T [] (λ j ψ j = 0(j = from above determiat represetatio By the -times iteratio of T [] the -fold DT has followig form ( ( 0 a T [] = T [] (λ; λ λ = λ b ( a + λ b + λ + + b b a b 0 λ + a 0 b b a b λ + a 0 b 0 0 b λ ψ ψ λ ψ ψ a ( which ca be determied by solvig the algebraic equatios T [] ψ j λ=λj = 0 j = Thus we get followig theorem Theorem The determiat represetatio of the -fold DT is expressed by (T [] = (T [] = (T [] = (T [] = T [] = W ( (T [] (T [] (T [] (T [] (3 0 λ 0 λ 0 λ 0 λ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ f ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ 0 0 λ 0 λ 0 λ 0 ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ f ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ 0 λ 0 λ 0 λ 0 λ 0 ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ f ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ λ 0 λ 0 λ 0 λ 0 ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ f ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ

4 4 Commuicatios i Theoretical Physics Vol 7 ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ f ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ W = λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ Corollary The -fold DT T [] geerates two ew solutios from iitial seed solutios I ad V amely I [] = I + b a 0 (4 V [] = b V a 0 + (5 b = W a0 = W b0 + = E EW a0 + = E EW ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ = λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ f ψ λ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ = ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ λ ψ The expressios of I [] give by Eqs (4 ad (5 are cosistet with results of Ref [7] It ca be see that T [] ad V [] is completely ew which is a coveiet tool = T [] (λ j ψ j for to geerate ozero eigefuctios ψ [] j j > which are elemetary bricks to costruct solito surfaces [37] 3 The -Degeerate Solutios I the last sectio ew solutios I [] ad V [] geerated by the T [] are give by the determiats ivolved with eigevalue λ j (j = ad their eigefuctios Settig I = V = 0 i Lax equatios Eqs d (3 it is straightforward to get -fold eigefuctios ( ( ψj λ j e λ jt ψ j = = j = (6 ψ j λ j e t/λ j Takig I = V = 0 ad above eigefuctios back ito Corollary it yields -solitos I [] = V [] = E EW (7 The aim of this sectio is to get -degeerate solitos by takig a limit λ j λ i -solitos Settig = Eq (7 produces two sigle solitos I [] e α = e α cosh(t sih α + α + α (8 V [] e α = e α cosh(t sih α + α + α (9 α = l λ There are two kids of solito i Eqs (8 ad (9 (i Whe λ > amely α > 0 the I [] is a bright solito (Fig (a ad V [] is a dark solito (Fig (b (ii Whe 0 < λ < amely α < 0 the I [] is a dark solito ad V [] is a bright solito The speed of two solitos is α /sih α The trajec-

5 o Commuicatios i Theoretical Physics 5 tory of I [] is a lie L 0 : t sih α + α / + α = 0 i the ( t-plae while L 0 : t sih α + α + α = 0 for the V [] The height of I [] ad V [] are ( e α / e α ad ( e α / e α respectively which ca be cofirmed by Fig with λ = 3 Fig The amplitudes of sigle solitos I [] ad V [] with λ = 3 (a The amplitude is 4/3 for I [] ; (b The amplitude is 4/3 for V [] I [] Settig = Eq (7 geerates two -solitos: ψ ψ λ ψ λ ψ λ ψ λ ψ λ ψ ψ ψ ψ λ ψ λ ψ = b λ a 0 = ψ λ ψ λ ψ ψ λ ψ ψ λ ψ λ ψ (0 ψ λ ψ λ ψ λ ψ λ ψ ψ λ ψ λ ψ ψ λ ψ λ ψ λ ψ V [] = b 0 + = EW Eψ λ Eψ λ Eψ λ Eψ λ Eψ Eψ λ Eψ λ Eψ Eψ λ Eψ λ Eψ λ Eψ λ Eψ Eψ λ Eψ λ Eψ ( I [] is plotted i Fig which icludes three cases: (i Two bright solitos (ii Oe bright solito ad oe dark solito (iii Two dark solitos We omit the similar profiles of V [] i order to keep a compact form of this paper It is straightforward to see from above formulas that I [] ad V [] become idetermiate form 0/0 whe eigevalues are degeerate ie λ λ This fact is also true for -soltios give by Eq (7 So it is iterestig to get -degeerate solito from a idetermiate form geerated by takig a limit λ j λ i -solito This ca be calculated by settig λ j = λ + ϵ ad usig higher-order Taylor expasio i Eq (7 Thus we reach followig Corollary Fig (Color oliethe discrete profiles of two-solito solutio I [] i the ( t-plae with discrete variable [ 5 5] (a Two bright soltios α = α = 5 (b Oe bright ad oe dark solitos α = α = 3 (c Two dark solitos α = 5 α = α j = l λ j j = Corollary The -degegrate solitos of Eq ( are expressed by I [] -ds = b a 0 = ( V [] -ds = b0 + = E EW (3

6 6 Commuicatios i Theoretical Physics Vol 7 ( ( i ij = ( ( i ij = ϵ=0 ϵ i ( ij (λ + ϵ ( ij (λ + ϵ ϵ=0 ϵ i ( ( ad i = [(i + /] [i] defie the floor fuctio of i Settig = Corollary yields a -degeerate solito I [] -ds which is plotted i Fig 3 ote that Fig 3(b is a desity plot of I [] -ds by usig cotiuous variable i order to get a good visibility ad all followig desity plots are geerated by this way To make a compact form of this paper profiles of V [] -ds are omitted Fig 3 (Color olie The discrete profile of a two-degeerate solutio I [] plot (b with α = -ds i the ( t-plae (a ad its desity Fig 4 (Color olie A sketchy demostratio of the limit λ λ i a two-solito I [] (desity plot with a = 0 c = / λ = i/5 s = 0 ad α = From the left to the right α = 3 5 ote that λ j = e α j (j = (c looks like Fig 3(b very much We are ow i a positio to demostrate ituitively the limit of degeeratio by a graphical way based o aalytical solutios I [] ad I [] -ds ie a -solito approaches to a -degeerate solito by λ λ ote that we set α j = l λ j (j = ad the use α α i order to show clearly this limit process It is see from Fig 4 that the trajectory (desity plot of a -solito approaches to the trajectory (Fig 3(b of a -degeerate solito whe α goes to α which shows vividly the limit of a -solito to a -degeerate solito Figure 4(c is a good approximatio of Fig 3(b although α = α α = 0 is ot very small Moreover whe t it is iterestig to fid i Fig 4 that the excellet agreemet betwee the exact trajectories (desity plots ad two approximate orbits amely L : t sih α + α + α + l(t = 0 L : t sih α + α + α l(t = 0 which shows the validity of the approximate orbits (black lies A remarkable feature of two approximate orbits is that there is a time-depedet phase shift amely l(t We are ot able to decompose properly a - degeerate solito ito two sigle solitos ulike we have obtaied a excellet decompositio for the real mkdv ad complex mkdv i Refs [ 3] because the - degeerate solito I [] -ds icludes a mixed combiatio of oe bright solito ad oe dark solito ad there exists a trasitio from a bright solito to a dark solito (or a

7 o Commuicatios i Theoretical Physics 7 reverse process alog the time evolutio retai this idea to fid above orbits But we still Fig 5 (Color olie The exact trajectories (desity plots red ad approximate orbits of a -degeerate solito I [] -ds The approximate orbits L (dashed lieblack ad L (solid lieblack i the ( t-plae with α = whe [ 0 0] From Fig 6 the excellet agreemet with exact trajectories(desity plots ad three approximate orbits (black lies for 3-degeerate solito amely ˆL : t sih α + α + α + 3 l(t = 0 ˆL : t sih α + α + α 3 l(t = 0 ˆL 3 : t sih α + α + α = 0 ote that there does ot exist phase shift for the solito propagatig alog ˆL 3 Fig 6 (Color olie The exact trajectories (desity plotsred ad approximate orbits of a 3-degeerate solito I [3] -ds with α = whe [ 5 5] The approximate orbits ˆL (dashed lieblack ˆL (solid lieblack ad ˆL 3 (dot lieblack i the ( t-plae 4 Summary ad Discussio I this article a determiat represetatio of the - fold Darboux trasformatio T [] of the oliear selfdual etwork equatio is give i Theorem It is easy to verify T [] (λ j ψ j = 0 (j = which shows ψ j (j = is the kerel of T [] Usig this represetatio the -solito solutios are obtaied i Eq (7 A geeral form of the -degeerate solitos is give i Corollary which is obtaied by a limit λ j λ ad by usig higher-order Taylor expasio from the -solitos For -degeerate ad 3-degeerate solitos approximate orbits (L i ˆL j (i = ; j = 3 are give aalytically which provide excellet fit of exact trajectories (desity plots see Figs 5 ad 6 A remarkable feature of approximate orbits i the degeerate solitos is that there is a time-depedet phase shift amely l(t If we compare our results with the work i Ref [7] of the self-dual etwork equatio our results have the followig advatages ad iovatio poits (i The determiat expressio T [] is give for the first time (ii The oe-solito solutio is aalyzed to classify the bright ad dark solito ad its height speed ad trajectory are preseted aalytically (iii The -degeerate solutios are give by determiats The approximate orbits of -degeerate ad 3- degeerate solitos ad the time-depedet phase shift are preseted i aalytical way Further by comparig T [] with the determiat represetatio T [][38] for the -fold DT i a cotiuous system it has followig differeces: (i The former has a differet form as a polyomial of λ due to the ew relatios of matrix coefficiets see Eq ( (ii we just replace oe colum i W to get ew solutios by oe same colum vector i former see Eqs (4 ad (5 while two colum vectors are eeded for a cotiuous system i Ref [38] Thus it is highly otrivial to exted determiat represetatio of -fold DT to discrete system Fially the explicit form of ψ [] j = T [] (λ j ψ j (j > ca be give aalytically by usig Theorem ad thus provides a coveiet tool to study the discrete solito surfaces ad the dyamical evolutio of ψ [] j i the ear future Ackowledgmets We thak Prof D J Zhag ad Dr X Y We for may helpful suggestios o this paper Refereces [] R Hirota J Phys Soc Jp 43 ( [] R Hirota J Phys Soc Jp 43 ( [3] R Hirota J Phys Soc Jp 43 ( [4] R Hirota J Phys Soc Jp 46 (979 3 [5] R Hirota J Phys Soc Jp 45 (978 3

8 8 Commuicatios i Theoretical Physics Vol 7 [6] B Kibler J Fatome C Fiot et al at Phys 6 ( [7] Y Ohta ad J K Yag J Phys A: Math Theor 47 ( [8] X Y We Commu Theor Phys 66 (06 9 [9] R Hirota J Phys Soc Jp 35 ( [0] M J Ablowitz ad J F Ladik J Math Phys 6 ( [] P G Kevrekidis The Discrete oliear Schrödiger Equatio Spriger Berli (009 [] D J Zhag J Ji ad X L Su Mod Phys Lett B 3 ( [3] L J og D J Zhag Y Shi et al Chi Phys Lett 30 ( [4] M Li J J Shuai ad T Xu Appl Math Lett 83 (08 0 [5] M J Ablowitz ad P A Clarkso oliear Evolutio Equatios ad Iverse Scatterig Cambridge Uiversity Press Cambridge (99 [6] Y Matsuo Biliear Trasformatio Method Burligto MA: Elsevier (984 [7] V B Matveev ad M A Salle Darboux Trasformatios ad Solitos Spriger-Verlag Berli (99 [8] V B Matveev Lett Math Phys 3 (979 3 [9] X G Geg Acta Math Sci 9 (989 [0] A Pickerig H Q Zhao ad Z Zhu Proc R Soc A 47 ( [] L H Wag J S He H Xu et al Phys Rev E 95 ( [] Q X Xig Z W Wu ad J S He oliear Dy 89 (07 99 [3] W Liu Y S Zhag ad J S He Wave Radom Complex 8 (08 03 [4] J S He H R Zhag L H Wag et al Phys Rev E 87 ( [5] Y S Zhag L J Guo J S He et al Lett Math Phys 05 ( [6] J S He S W Xu K Porsezia et al Phys Rev E 93 ( [7] X Y We J Phys Soc Jp 8 ( [8] V E Zakharov ad A B Shabat Sov Phys JETP 34 (97 6 [9] V B Matveev Phys Lett A 66 (99 05 [30] V B Matveev Phys Lett A 66 (99 09 [3] A A Stahlofe A Phys (Berli 504 ( [3] H Maisch ad A A Stahlofe Phys Scripta 5 (995 8 [33] R Beutler J Math Phys 34 ( [34] A A Stahlofe ad V B Matveev J Phys A: Math Ge 8 ( [35] D J Zhag S L Zhao Y Y Su et al Rev Math Phys 6 ( [36] V B Matveev Theor Math Phys 3 ( [37] A Sym Solito Surfaces ad Their Applicatios i Geometric Aspects fo the Eistei Equatio ad Itegrable Systems ed R Martii Spriger-Verlag ew York ( [38] J S He L Zhag Y Cheg et al Sci Chia Ser A 49 (