Commun. Theor. Phys. 55 011 960 964 Vol. 55 No. 6 June 15 011 A Limi Symmery o Modiied KdV Equaion and Is Applicaions ZHANG Jian-Bing Ï 1 JI Jie SHEN Qing ã 3 and ZHANG Da-Jun 3 1 School o Mahemaical Sciences Xuzhou Normal Universiy Xuzhou 1116 China College o Saisics and Mahemaics Zheiang Gongshang Universiy Hangzhou 310018 China 3 Deparmen o Mahemaics Shanghai Universiy Shanghai 00444 China Received Sepember 16 010 Absrac In his leer we consider a limi symmery o he modiied KdV equaion and is applicaion. The similariy reducion leads o limi soluions o he modiied KdV equaion. Besides a modiied KdV equaion wih new sel-consisen sources is obained and is soluions are derived. PACS numbers: 0.30.Ik 05.45.Yv Key words: symmery he mkdv equaion symmery reducion sel-consisen source Hiroa s mehod 1 Inroducion I is well known ha solion soluions can be derived by means o variey o approaches such as he Inverse Scaering Transormaion IST Darbou ransormaion Bäcklund ransormaion algebraic geomery approach bilinear mehod and so on. From he viewpoin o he IST N solions are ideniied by N disinc eigenvalues or in oher words N disinc simple poles {k } o ransparen coeicien 1/ak. In he symmery approach [1] he classical N-solion soluion is relaed o a similariy reducion o squared-eigenuncion symmeries. [3] In he leer we consider a new symmery or he modiied KdV mkdv equaion. The symmery is relaed o he known squared eigenuncion symmery by a limi procedure and he group invarian soluion rom he relaed λ u u λ λ is he specral parameer. I is known ha σ 1 φ 3 is a symmery o he mkdv equaion i φ 1 and saisy. According o he lineariy o linear equaion which he symmery saisies his symmery ogeher wih anoher symmery u inroduces a combined symmery λ u u λ similariy reducion is a double-pole soluion Res. [4 5] which is a limi soluion. [6] Besides squared-eigenuncion symmeries are relaed o sel-consisen sources o solion equaions. [78] The limi soluion hen leads o an mkdv equaion wih new sel-consisen sources. We use Hiroa s bilinear mehod o ge he soluions or his equaion. The leer is organized as ollows. In Sec. we discuss he limi symmery. In Sec. 3 he relaed similariy reducion is obained and solved. Secion 4 invesigaes he mkdv equaion wih new sources and is soluions. A Limi Symmery o mkdv Equaion The mkdv equaion is wih La pair u 6u u u 0 1 4λ 3 λu 4λ u λu u u 3 4λ u λu u u 3 4λ 3 λu ˆσ u. 4 Thus we obain a symmery consrain u 5 4λ 3 λ u 4λ u λ u u u 3 4λ u λ u u u 3 4λ 3 λ u. 6 As in Re. [] he above epression can be derived rom he Gel and Levian Machenko equaion in he IST procedure [9] also acs as a saring poin o nonlinearizaion o La pair. [10] Suppored by he Naional Naural Science Foundaion o China under Gran Nos. 10871165 and 1096036 he Educaion Deparmen under Gran No. Y00906909 and he Naural Science Foundaion o under Gran No. Y610016 o Zheiang Province Corresponding auhor E-mail: ianbingzhang@yahoo.cn c 011 Chinese Physical Sociey and IOP Publishing Ld hp://www.iop.org/ej/ournal/cp hp://cp.ip.ac.cn
No. 6 Communicaions in Theoreical Physics 961 In he leer we consider σ φ ψ ψ 7 φ is he wave uncion in Eq. 6 and ψ λ u ψ 1 0 ψ u λ ψ 0 1 ψ 4λ 3 λ u 4λ u λ u u u 3 ψ ψ 4λ u λ u u u 3 4λ 3 λ u ψ 1λ u 8λ u u 8λ u u 1λ. 8 u By direc veriicaion we ind σ saisies σ 6u σ 1uu σ σ 9 which means σ is a symmery o he mkdv equaion as well. Since Eq. 8 can be hough as a derivaive o Eq. 6 we call σ a limi symmery. For anoher in Sec. 3 we will see soluions generaed rom σ by a similariy reducion can be looked as limi soluions. 3 Similariy Reducion and Soluions We consider he combined symmery σ u φ ψ ψ 10 φ solve Eq. 6 and ψ ψ solve he auiliary sysem Eq. 8. Then he consrain σ 0 yields u φ ψ ψ. 11 Since when φ k ψ k k 1 or saisy Eqs. 6 and 8 u deined by Eq. 11 saisies he mkdv equaion 1 auomaically so he consrain sysem is reduced o u φ ψ ψ φ λ φ u uφ λ φ 4λ 3 λ u φ 4λ u λ u u u 3 4λ u λ u u u 3 φ 4λ 3 λ u ψ λ ψ uψ φ ψ uψ λ ψ ψ 4λ 3 λ u ψ 4λ u λ u u u 3 ψ 1λ u φ 8λ u u ψ 4λ u λ u u u 3 ψ 4λ 3 λ u ψ 1λ u 8λ u u φ. 1 I can be wrien ino bilinear orms as wih λ k D i N D 0 D ḡ k g D h k h g ḡ h h g D D 3 3k D g 0 D D 3 3k D h 6k D g 0 13 by he dependen variable ransormaions u i ln φ ḡ g i ḡ g ψ h h ψ i h h 14 ḡ h are he comple conugaes o g h and D is he well-known Hiroa s bilinear operaor deined by [11] D m Da n b m n s m y n a s yb s y s0y0 m n 0 1... Ne we epand g h as 1 l ε l g g l1 ε l1 h l1 l1 l1 h l1 ε l1 15
96 Communicaions in Theoreical Physics Vol. 55 and subsiue hem ino Eq. 13. When N 1 he epansions can be runcaed by aking l g l1 1 h l1 1 0 or l 3 hen we have i 1k1 1 e ξ1 k 1 4 1 e 4ξ1 g 1 1 k 1 e ξ1 g 3 1 i k 1 e 3ξ1 1 h 1 1 k 1 e ξ1 1k 1 h 3 1 i k 1 3k 1 1 4k 1 e 3ξ1 16a 16b 16c l g l1 1 h l1 1 0 l 3 16d ξ 1 k 1 4k 3 1 ξ 0 1 16e wih real parameers k 1 e ξ0 1. We invesigae dynamics o he above soluion i.e. 1 i 1k1 1 e ξ1 1 k 1 e 4ξ1 u i ln arcan 1k 1 1/k 1 e ξ1 1/ 17 eξ1 we have aken ε 1 in Eq. 15. As depiced in Fig. 1a he soluion is smooh and i has wo normal solions wih consan-ampliude k 1. To realize he asympoic behaviors o such a soluion we pu i in he ollowing moving coordinae rame Fig. 1b X 4k1. 18 By means o asympoic analysis we ind ha he our branches in Fig. 1b asympoically are k 1 > 0 X 1 k 1 ln ln3k 3 1 X X 1 k 1 ln ln3k 3 1 X X 1 k 1 ln ln3k 3 1 X X 1 k 1 ln ln3k 3 1 X 19 and he ampliude o each separaed solion is k 1. Ne le us invesigae he relaion beween he obained soluion 16 and he known -solion soluion o he mkdv equaion u i ln 1 ie ξ1 e ξ k1 k k 1 k e ξ 1ξ 0 ξ 1 is deined as Eq. 16e. We subsiue α 1 e ξ0 1 β 1 k 1 k 1 k and α 1 e ξ 0 1 β 1 k k k 1 or e ξ0 1 and e ξ0 respecively α 1 is a real consan and β k is a diereniable uncion o k. So we can wrie Eq. 0 as e ξ1 e ξ α1 1 iα 1 e ξ 1ξ 1 k 1 k k 1 k wih new ξ ξ k 4k 3 β k ξ 0 1. Taking k k 1 and using L Hospial rule Eq. 1 goes o 1iα 1 1k1 k1 β 1 k 1 e ξ1 α 1 4k1 e 4ξ1 3 which equals o Eq. 17 when α 1 1/ β 1 k 1 1/lnk 1. Tha implies soluions derived by he symmery consrain Eq. 11 are limi soluions. Fig. 1 Plos or u given by Eq. 7 wih k 1 1.5 ξ 0 1 0. a Shape o u or 80; b Plo 3D picure or u. 4 The mkdv Equaion wih New Sources Squared eigenuncions can be used o generae sources or solion equaions. [781] The equaions u u 6u u φ 0 φ λ φ u
No. 6 Communicaions in Theoreical Physics 963 uφ λ 4 are called he mkdv equaion wih sel-consisen sources mkdvescs. By he limi symmery 7 we have u u 6u u φ ψ ψ 0 φ λ φ u uφ λ ψ λ ψ uψ φ ψ uψ λ ψ 5 {k } N are disinc real numbers. This sysem is La inegrable wih he La pair λ u u λ A B C A A 4λu 8λ 3 1 1 u 1 1 u B 8λ u 4λu u 4u 3 1 λ ψ φ ψ λφ φ λ λ λ λ λ ψ φ ψ λφ φ λ λ λ λ λ φ ψ φ λ λ λ λ λ λ λ C 8λ u 4λu u 4u 3 1 λ ψ φ λ λ λ λ λ λ λ λ ψ φ λ λ λ λ λ λ λ. λ φ ψ φ λ λ λ λ λ λ λ To derive his we use he ac φ L φ λ ψ ψ L λ ψ ψ φ L φ λ φ ψ L φ ψ λ φ ψ φ ψ φ φ u 1 u u 1 u L u 1 u u 1 u and φ ψ saisy Eqs. 6 and 8. Sill using he ransormaion 14 one can wrie Eq. 5 ino wih λ k D D 3 i ḡ h g h D 0 D ḡ k g D h k h g. 6 This can be solved as in Sec. 3. When N 1 we have i 1k1e ξ1 4 1 e 4ξ1 7a g 1 1 β 1 e ξ1 g 3 1 i 1 β 1 e 3ξ1 7b h 1 1 β 1 e ξ1 1k1 h 3 1 i β 1 3k 1 e 3ξ1 g 1 1 h 1 1 0 3 ξ k 4k 3 0 7c β 1 zdz ξ 0 7d k e ξ0 are real parameers and 0 β 1zdz is arbirary -dependen uncion. Thus a soluion or Eq. 5 is 1 i 1k1 1 e eξ1 4ξ1 u i ln arcan 1k1 e ξ1 1/ eξ1 8 we have aken ε 1 in Eq. 15. We depic i in Fig..
964 Communicaions in Theoreical Physics Vol. 55 Fig. Plos or u given by Eq. 8 k 1 0.5 ξ 0 1 0 β 1z z. a Shape o u or 15. b Plo 3D picure or u. 5 Conclusions We have considered he limi squared-eigenuncion symmery N φ ψ ψ or he mkdv equaion and is applicaions. The obained soluion N 1 case relaed o he similariy reducion is shown o be a limi soluion o -solion soluion o he mkdv equaion. Dynamics is analyzed. Asympoically he soluion consiss o wo separaed solions raveling wih logarihmic raecories. Besides we give an mkdvescs o which he sources are relaed o he limi symmery. The equaion is La inegrable and we also derive a soluion via Hiroa mehod. Reerences [1] P. Olver Applicaions o Lie Groups o Dierenial Equaions nd ed. Springer-Verlag New York 1993. [] C. Gardner J. Greene M. Kruskal and R. Miura Commun. Pure Appl. Mah. 7 1974 97. [3] J.B. Zhang D.J. Zhang and D.Y. Chen Commun. Theor. Phys. 53 010 11. [4] M. Wadai and K. Ohkuma J. Phys. Soc. Jpn. 51 198 09. [5] D.Y. Chen D.J. Zhang and S.F. Deng J. Phys. Soc. Jpn. 71 00 658. [6] M. Takahashi and K. Konno J. Phys. Soc. Jpn. 58 1989 3505. [7] Y.B. Zeng and Y.S. Li J. Phys. A: Mah. Gen. 6 1993 73. [8] R.L. Lin Y.B. Zeng and W.X. Ma Physica A 91 001 87. [9] C.W. Cao Sci. China. 33A 1990 58. [10] C.W. Cao and X.G. Geng in Nonlinear Physics Research Repors in Physics eds. by C.H. Gu Y.S. Li and G.Z. Tu Springer Berlin 1990 68. [11] R. Hiroa Phys. Rev. Le. 7 1971 119. [1] D.J. Zhang J.B. Zhang and Q. Shen Teore. Ma. Fiz. 163 010 77.