1National Advanced School of Engineering, University of Yaounde I, P.O. Box. 8390, Cameroon 2Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box. 812, Cameroon 3Department of Mathematics, Higher Teacher's Training College, University of Maroua, P.O. Box. 46, Cameroon 4Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon
Prolongation Structure Analysis of a Coupled Dispersionless System
1National Advanced School of Engineering, University of Yaounde I, P.O. Box. 8390, Cameroon 2Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box. 812, Cameroon 3Department of Mathematics, Higher Teacher's Training College, University of Maroua, P.O. Box. 46, Cameroon 4Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon
摘要We address the problem of integrability of a coupled dispersionless system recently introduced by Zhaqilao, Zhao and Li [Chin. Phys. B 18 (2009) 1780] which physically describes the propagation of electromagnetic fields within an optical nonlinear medium, but also arrives in the physical description of a charged object dynamics in an external magnetic field. Following the prolongation structure analysis developed by Wahlquist and Estabrook, we derive a more general form of Lax pairs of the previous coupled dispersionless system and its concrete non-Abelian Lie algebra resorting to a hidden symmetry. Also, we construct the Bäcklund transformation of the system using the Riccati form of the linear eigenvalue problem.
Abstract：We address the problem of integrability of a coupled dispersionless system recently introduced by Zhaqilao, Zhao and Li [Chin. Phys. B 18 (2009) 1780] which physically describes the propagation of electromagnetic fields within an optical nonlinear medium, but also arrives in the physical description of a charged object dynamics in an external magnetic field. Following the prolongation structure analysis developed by Wahlquist and Estabrook, we derive a more general form of Lax pairs of the previous coupled dispersionless system and its concrete non-Abelian Lie algebra resorting to a hidden symmetry. Also, we construct the Bäcklund transformation of the system using the Riccati form of the linear eigenvalue problem.
[1] Kodama Y 1988 Phys. Lett. A 129 223
[2] Takasaki K and Takebe T 1992 Int. J. Mod. Phys. 7 889
[3] Krichever M 1994 Commun. Pure Appl. Math. 47 437
[4] Zhao X Q and Lu J F 1999 J. Phys. Soc. Jpn. 68 2151
[5] Zhaqilao, Zhao Y L and Li Z B 2009 Chin. Phys. B 18 1780
[6] Konno K 1995 Appl. Anal. 57 209
[7] Kakuhata H and Konno K 1997 J. Phys. A 30 L401
[8] Kakuhata H and Konno K 1996 J. Phys. Soc. Jpn. 65 340
[9] Lamb G L Jr 1971 Rev. Mod. Phys. 43 99
[10] Ablowitz M J, Kaup D J, Newell A C and Segur H 1973 Phys. Rev. Lett. 31 125
[11] Wahlquist H D and Estabrook F B 1975 J. Math. Phys. 16 1
[12] Bracken P 2010 J. Math. Phys. 2010 51 113502
[13] Bracken P 2010 J. Math. Stat. 2010 6 125
[14] Cao Y H and Wang D S 2010 Commun. Nonlinear Sci. Numer. Sumulat. 15 2344
[15] Wang D S 2010 Nonlinear Analysis 73 270
[16] Wang D S 2010 Appl. Math. Lett. 23 665
[17] Wang D S 2010 Appl. Math. Comput. 216 1349
[18] Wu G C 2009 Chaos Solitons Fractals 42 408
[19] Duan X J, Deng M, Zhao W Z and Wu K 2007 J. Phys. A 40 3831
[20] Zhai Y, Albeverio S, Zhao W Z and Wu K 2006 J. Phys. A 39 2117
[21] Ming D and Li L M 2010 Commun. Theor. Phys. 53 218
[22] Tao X and Li L M 2008 Commun. Theor. Phys. 50 565
[23] Bouetou B T, Souleymanou A, Kuetche K V, Mouna F and Kofane T C 2011 J. Math. Anal. Appl. 377 269
[24] Kuetche K V, Bouetou B T and Kofane T C 2010 Handbook of Solitons: Research, Technology and Applications , (New-York: Nova Science Publishers)
[25] Cartan E 1945 Les Systèmes Différentiels Extérieurs et Leurs Applications Géométriques (Paris: Hermann)
[26] Harrison B K and Estabrook F B 1971 J. Math. Phys. 12 653
[27] Choquet Y B 1968 Géométrie Différentielle et Systèmes Extérieurs (Paris: Dunod)
[28] Slebodzinski W 1970 Exterior Forms and Applications , (Warsaw: Polish Scientific Publishers)
[29] Flanders H 1963 Differential Forms (New York: Academic)
[30] Leo M, Leo R A, Martina L, Pirani F A E and Soliani G 1981 Physica D 4 105
[31] Leo M, Leo R A, Soliani G and Martina L 1982 Phys. Rev. D 26 809
[32] Morris H C 1976 J. Math. Phys. 17 1867
[33] Lakshmanan M 1978 J. Math. Phys. 20 1667
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