Two \((2+1)\)-dimensional soliton equations and their quasi-periodic solutions. (English) Zbl 1080.35110
Summary: Two \((2 + 1)\)-dimensional soliton equations and their decomposition into the mixed \((1+1)\)-dimensional soliton equations are proposed. With the help of nonlinearization approach, the Lenard spectral problem related to the mixed soliton hierarchy is turned into a completely integrable Hamiltonian system with a Lie-Poisson structure on the Poisson manifold \(\mathbb R^{3N}\). The Abel-Jacobi coordinates are introduced to straighten out the Hamiltonian flows. Based on the decomposition and the algebraic curve theory, explicit quasi-periodic solutions for the \((1+1)\)- and \((2+1)\)-dimensional soliton equations are obtained.
MSC:
| 35Q51 | Soliton equations |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
References:
| [1] | Bonopelchenko, B.; Sidorenko, J.; Strampp, W., Phys Lett A, 157, 157 (1991) |
| [2] | Cheng, Y.; Li, Y. S., Phys Lett A, 157, 22 (1991) |
| [3] | Cheng, Y.; Li, Y. S., J Phys A, 25, 419 (1992) · Zbl 0749.35037 |
| [4] | Cao, C. W.; Wu, Y. T.; Geng, X. G., J Math Phys, 40, 3948 (1999) · Zbl 0947.35138 |
| [5] | Cao, C. W.; Geng, X. G.; Wang, H. Y., J Math Phys, 48, 621 (2002) |
| [6] | Geng, X. G.; Wu, Y. T., J Math Phys, 40, 2971 (1999) · Zbl 0944.35085 |
| [7] | Geng, X. G.; Wu, Y. T.; Cao, C. W., J Phys A: Math Gen, 32, 3733 (1999) · Zbl 0941.35090 |
| [8] | Cao, C. W.; Geng, X. G.; Wu, Y. T., J Phys A: Math Gen, 32, 8059 (1999) · Zbl 0977.37036 |
| [9] | Geng, X. G.; Cao, C. W., Nonlinearity, 14, 1433 (2001) · Zbl 1160.37405 |
| [10] | Konopelchenko, B.; Dubrovsky, V., Phys Lett A, 102, 15 (1984) |
| [11] | Konopelchenko, B., Solitons in multidimensions, inverse spectral transform method (1993), World Scientific: World Scientific Singapore · Zbl 0836.35002 |
| [12] | Bogoyavlenskii, O. I., Usp MAt Nauk, 45, 17 (1990) |
| [13] | Bogoyavlenskii, O. I., Izv Akad Nauk SSSR Ser Mat, 54, 1123 (1989) |
| [14] | Li, Y. S.; Zhang, Y. J., J Phys A: Math Gen, 26, 943 (1993) |
| [15] | Geng, X. G., J Phys A: Gen, 36, 2289 (2003) · Zbl 1039.37061 |
| [16] | Jiang, Z. H.; Bullough, R. K., Phys Lett A, 190, 249 (1994) · Zbl 0959.35512 |
| [17] | Qin, Z. Y.; Zhou, R. G., Chaos, Solitons & Fractals, 21, 311 (2004) · Zbl 1049.35152 |
| [18] | Wadati, M.; Konno, K.; Ichikawa, Y. H., J Phys Soc Jpn, 46, 1965 (1979) · Zbl 1334.81106 |
| [19] | Ablowitz, M. J.; Clarkson, P. A., Solitons, nonlinear evolution equations and inverse scattering (1991), Cambridge University Press · Zbl 0762.35001 |
| [20] | Abraham, R.; Marsden, J. E., Foundations of mechanics (1978), Benjamin/Cummings: Benjamin/Cummings London · Zbl 0393.70001 |
| [21] | Sattinger, D. H.; Weaver, O. L., Lie groups and algebras with applications to physics, geometry and mechanics (1986), Springer · Zbl 0595.22017 |
| [22] | Olver, P. J., Applications of Lie groups to differential equations (1986), Springer: Springer New York · Zbl 0656.58039 |
| [23] | Marsden, J. E., Introduction to mechanics and symmetry (1994), Springer: Springer New York · Zbl 0811.70002 |
| [24] | Weinstein, A., J Diff Geom, 18, 523 (1983) · Zbl 0524.58011 |
| [25] | Du, D. L.; Cao, C. W., Phys Lett A, 278, 209 (2001) · Zbl 0972.37050 |
| [26] | Du, D. L., Physica A, 303, 439 (2002) |
| [27] | Wu, Y. T.; Du, D. L., J Math Phys, 41, 5832 (2000) · Zbl 1026.37057 |
| [28] | Siegel, C. L., Topics in complex function theory, vol. 2 (1971), Wiley: Wiley New York · Zbl 0211.10501 |
| [29] | Griffiths, P.; Harris, J., Principles of algebraic geometry (1999), Wiley: Wiley New York |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
