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Bäcklund transformation, Lax pair, and solutions for the Caudrey-Dodd-Gibbon equation. (English) Zbl 1314.37050
Summary: By using Bell polynomials and symbolic computation, we investigate the Caudrey-Dodd-Gibbon equation analytically. Through a generalization of Bells polynomials, its bilinear form is derived, based on which, the periodic wave solution and soliton solutions are presented. And the soliton solutions with graphic analysis are also given. Furthermore, Bäcklund transformation and Lax pair are derived via the Bells exponential polynomials. Finally, the Ablowitz-Kaup-Newell-Segur system is constructed.
©2011 American Institute of Physics

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35C08 Soliton solutions
33E10 Lamé, Mathieu, and spheroidal wave functions
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References:
[1] Bullough, R. K.; Caudrey, P. J., Solitons, (1980), Springer-Verlag: Springer-Verlag, Berlin
[2] Tian, B.; Gao, Y. T., Phys. Plasmas, 12, 054701, (2005), 10.1063/1.1885477; Tian, B.; Gao, Y. T., Phys. Plasmas, 12, 054701, (2005), 10.1016/j.physleta.2005.03.082; Tian, B.; Gao, Y. T., Phys. Plasmas, 12, 054701, (2005), 10.1016/j.physleta.2005.03.082; Tian, B.; Gao, Y. T., Phys. Plasmas, 12, 054701, (2005), 10.1016/j.physleta.2005.03.082; · Zbl 1145.35452
[3] Das, G.; Sarma, J., Phys. Plasmas, 6, 4394, (1999), 10.1063/1.873705; Das, G.; Sarma, J., Phys. Plasmas, 6, 4394, (1999), 10.1088/0305-4470/34/8/320; Das, G.; Sarma, J., Phys. Plasmas, 6, 4394, (1999), 10.1140/epjd/e2005-00036-6; Das, G.; Sarma, J., Phys. Plasmas, 6, 4394, (1999), 10.1016/j.physleta.2005.03.035; Das, G.; Sarma, J., Phys. Plasmas, 6, 4394, (1999), 10.1016/j.physleta.2005.03.035; · Zbl 1145.35451
[4] Barnett, M. P.; Capitani, J. F.; Von Zur Gathen, J.; Gerhard, J., Int. J. Quantum Chem., 100, 80, (2004), 10.1002/qua.v100:2; Barnett, M. P.; Capitani, J. F.; Von Zur Gathen, J.; Gerhard, J., Int. J. Quantum Chem., 100, 80, (2004), 10.1063/1.2363352; Barnett, M. P.; Capitani, J. F.; Von Zur Gathen, J.; Gerhard, J., Int. J. Quantum Chem., 100, 80, (2004), 10.1063/1.2363352;
[5] Matveev, V. B.; Salle, M. A., Darboux Transformation and Solitons, (1991), Springer-Verlag: Springer-Verlag, Berlin; Matveev, V. B.; Salle, M. A., Darboux Transformation and Solitons, (1991), Springer-Verlag: Springer-Verlag, Berlin, 10.1088/0305-4470/27/13/035; · Zbl 0842.35103
[6] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, (1991), Cambridge University Press: Cambridge University Press, New York; Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, (1991), Cambridge University Press: Cambridge University Press, New York, 10.1143/JPSJ.52.2642; · Zbl 0762.35001
[7] Wadati, M., J. Phys. Soc. Jpn., 38, 673, (1975), 10.1143/JPSJ.38.673; Wadati, M., J. Phys. Soc. Jpn., 38, 673, (1975), 10.1143/PTP.53.419; Wadati, M., J. Phys. Soc. Jpn., 38, 673, (1975), 10.1143/PTP.53.1652; · Zbl 1079.35505
[8] Tian, B.; Gao, Y. T., Phys. Plasmas, 12, 070703, (2005), 10.1063/1.1950120; Tian, B.; Gao, Y. T., Phys. Plasmas, 12, 070703, (2005), 10.1016/j.physleta.2006.11.019; Tian, B.; Gao, Y. T., Phys. Plasmas, 12, 070703, (2005), 10.1209/0295-5075/77/15001; · Zbl 1325.35192
[9] Hirota, R., The Direct Method in Soliton Theory, (2004), Cambridge University Press: Cambridge University Press, Cambridge; Hirota, R., The Direct Method in Soliton Theory, (2004), Cambridge University Press: Cambridge University Press, Cambridge, 10.1103/PhysRevLett.27.1192; · Zbl 1168.35423
[10] Hirota, R., Prog. Theor. Phys. Suppl., 52, 1498, (1974), 10.1143/PTP.52.1498; Hirota, R., Prog. Theor. Phys. Suppl., 52, 1498, (1974), 10.1143/JPSJ.41.2141; Hirota, R., Prog. Theor. Phys. Suppl., 52, 1498, (1974), 10.1143/PTPS.59.64;
[11] Ma, W. X.; You, Y. C., Trans. Am. Math. Soc., 357, 1753, (2005) · Zbl 1062.37077
[12] Hirota, R.; Satsuma, J., J. Phys. Soc. Jpn., 45, 1741, (1978)
[13] Caudrey, P. J.; Dodd, R. K.; Gibbon, J. D., Proc. R. Soc. London, Ser. A, 351, 407, (1976) · Zbl 0346.35024
[14] Wazwaz, A. M., Appl. Math. Comput., 203, 402, (2008) · Zbl 1185.65192
[15] Weiss, J., J. Math. Phys., 25, 13, (1984) · Zbl 0565.35094
[16] Aiyer, R. N.; Fuchssteiner, B.; Oevel, W., J. Phys. A, 19, 3755, (1986) · Zbl 0622.35067
[17] Fuchssteiner, B.; Oevel, W., J. Math. Phys., 23, 358, (1982) · Zbl 0489.35029
[18] Levin, D.; Ragnisco, O., Inverse Probl. Eng., 4, 815, (1988) · Zbl 0694.35211
[19] Ma, W. X.; Zhou, R. G.; Gao, L., Mod. Phys. Lett. A, 24, 1677, (2009) · Zbl 1168.35426
[20] Bell, E. T., Ann. Math., 35, 258, (1934) · Zbl 0009.21202
[21] Gilson, C.; Lambert, F.; Nimmo, J.; Willox, R., Proc. R. Soc. London, Ser. A, 452, 223, (1996) · Zbl 0868.35101
[22] Nakamura, A., J. Phys. Soc. Jpn., 47, 1701, (1979), 10.1143/JPSJ.47.1701; Nakamura, A., J. Phys. Soc. Jpn., 47, 1701, (1979), 10.1143/JPSJ.47.1701; · Zbl 1334.35006
[23] Lambert, F.; Springael, J., Chaos, Solitons Fractals, 12, 2821, (2001) · Zbl 1005.37043
[24] Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segure, H., Phys. Rev. Lett., 31, 125, (1973), 10.1103/PhysRevLett.31.125; Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segure, H., Phys. Rev. Lett., 31, 125, (1973), 10.1103/PhysRevLett.31.125; Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segure, H., Phys. Rev. Lett., 31, 125, (1973), 10.1103/PhysRevLett.31.125; · Zbl 1243.35143
[25] Matsuno, Y., Bilinear Transformation Method, (1984), Academic: Academic, New York · Zbl 0552.35001
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