Integrability of a differential-difference KP equation with self-consistent sources. (English) Zbl 1114.37039
Summary: We introduce a differential-difference KP equation with self-consistent sources
(D\(\Delta\)KPESCS) which is a generalization of the D\(\Delta\)KP equation. The integrability of the differential-difference equation is shown through bilinear transformation method and Wronskian technique: it possesses \(N\)-soliton solution expressed in terms of Casorati determinants, bilinear Bäcklund transformation and Lax pairs.
(D\(\Delta\)KPESCS) which is a generalization of the D\(\Delta\)KP equation. The integrability of the differential-difference equation is shown through bilinear transformation method and Wronskian technique: it possesses \(N\)-soliton solution expressed in terms of Casorati determinants, bilinear Bäcklund transformation and Lax pairs.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 35Q51 | Soliton equations |
| 35Q58 | Other completely integrable PDE (MSC2000) |
Keywords:
solitons; Bäcklund transformation; Lax pair; differential-difference KP equation with sourcesReferences:
| [1] | Claude, C.; Latifi, A.; Leon, J., Nonlinear resonant scattering and plasma instability: an integrable mode, J. Math. Phys., 32, 3321-3330 (1991) · Zbl 0744.76056 |
| [2] | Deng, S. F.; Chen, D. Y.; Zhang, D. J., The multisoliton solutions of the KP equation with self-consistent sources, J. Phys. Soc. Jpn., 72, 2184-2192 (2003) · Zbl 1056.35139 |
| [3] | Doktorov, E. V.; Shchesnovich, V. S., Nonlinear evolutions with singular dispersion laws associated with a quadratic bundle, Phys. Lett. A, 207, 153-158 (1995) · Zbl 1020.37525 |
| [4] | Doktorov, E. V.; Vlasov, R. A., Optical solitons in media with resonant and nonresonant self-focusing nonlinearities, Opt. Acta, 30, 223-232 (1983) |
| [5] | Freeman, N. C.; Nimmo, J. J.C., Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique, Phys. Lett. A, 95, 1-3 (1983) · Zbl 0588.35077 |
| [6] | Gegenhasi, X. B.; Hu, On an integrable differential-difference equation with a source, J. Nonlinear Math. Phys., 13, 2, 183-192 (2006) · Zbl 1110.35346 |
| [7] | R. Hirota, Direct Method in Soliton Theory [Edited and Translated by A. Nagai, J. Nimmo, C. Gilson] Cambridge University Press, 2004) (in English).; R. Hirota, Direct Method in Soliton Theory [Edited and Translated by A. Nagai, J. Nimmo, C. Gilson] Cambridge University Press, 2004) (in English). · Zbl 1099.35111 |
| [8] | Hu, X. B., The higher-order KdV equation with a source and nonlinear superposition formula, Chaos Solitons Fract., 7, 211-215 (1996) · Zbl 1080.35535 |
| [9] | Leon, J., Nonlinear evolutions with singular dispersion laws and forced systems, Phys. Lett. A, 144, 444-452 (1990) |
| [10] | Leon, J.; Latifi, A., Solution of an initial-boundary value problem for coupled nonlinear waves, J. Phys. A, 23, 1385-1403 (1990) · Zbl 0713.35084 |
| [11] | Lin, R. L.; Zeng, Y. B.; Ma, W. X., Solving the KdV hierarchy with self-consistent sources by inverse scattering method, Physica A, 291, 287 (2001) · Zbl 0972.35128 |
| [12] | Liu, X. J.; Zeng, Y. B., On the Toda lattice equation with self-consistent sources, J. Phys. A, 38, 8951-8965 (2005) · Zbl 1113.37053 |
| [13] | Ma, W. X., Soliton, Positon and Negaton Solutions to a Schrödinger Self-consistent Source Equation, J. Phys. Soc. Jpn., 72, 3017-3019 (2003) · Zbl 1133.81332 |
| [14] | Matsuno, Y., Kadomtsev-Petviashvili equation with a source and its soliton solutions, J. Phys. A, 23, 1235-L1239 (1990) · Zbl 0729.35116 |
| [15] | Mel’nikov, V. K., A direct method for deriving a multisoliton solution for the problem of interaction of waves on the x,y plane, Commun. Math. Phys., 112, 639-652 (1987) · Zbl 0647.35077 |
| [16] | Mel’nikov, V. K., Integration method of the Korteweg-de Vries equation with a self-consistent source, Phys. Lett. A, 133, 493-496 (1988) |
| [17] | Mel’nikov, V. K., Capture and confinement of solitons in nonlinear integrable systems, Commum. Math. Phys., 120, 451-468 (1989) · Zbl 0669.58035 |
| [18] | Mel’nikov, V. K., Interaction of solitary waves in the system described by the Kadomtsev-Petviashvili equation with a self-consistent source, Commun. Math. Phys., 126, 201-215 (1989) · Zbl 0682.76013 |
| [19] | Mel’nikov, V. K., New method for deriving nonlinear integrable systems, J. Math. Phys., 31, 1106-1113 (1990) · Zbl 0705.70003 |
| [20] | Mel’nikov, V. K., Integration of the nonlinear Schrödinger equation with a source, Inverse Probl., 8, 133-147 (1992) · Zbl 0752.35069 |
| [21] | Nimmo, J. J.C., Soliton solution of three differential-difference equations in Wronskian form, Phys. Lett. A, 99, 281-286 (1983) · Zbl 1168.34365 |
| [22] | Shchesnovich, V. S.; Doktorov, E. V., Modified Manakov system with self-consistent source, Phys. Lett. A, 213, 23-31 (1996) · Zbl 1073.35531 |
| [23] | Tamizhmani, T.; Kanaga, V. S.; Tamizhmani, K. M., Wronskian and rational solutions of the differential-difference KP equation, J. Phys. A., 31, 7627 (1998) · Zbl 0931.35154 |
| [24] | Tamizhmani, K. M.; Kanagavel, S.; Grammaticos, B.; Ramani, A., Singularity structure and algebraic properties of the differential-difference Kadomtsev-Petviashvili equation, Chaos Solitons Fractals, 11, 9, 1423-1431 (2000) · Zbl 1121.37318 |
| [25] | H.Y. Wang, X.B. Hu, Gegenhasi, Toda lattice equation with self-consistent sources: Casoratian type solution, bilinear Bäcklund transformation and Lax pair, J. Comp. Appl. Math., doi:10.1016/j.cam.2005.08.052.; H.Y. Wang, X.B. Hu, Gegenhasi, Toda lattice equation with self-consistent sources: Casoratian type solution, bilinear Bäcklund transformation and Lax pair, J. Comp. Appl. Math., doi:10.1016/j.cam.2005.08.052. · Zbl 1172.37028 |
| [26] | Xiao, T.; Zeng, Y. B., Generalized Darboux transformations for the KP equation with self-consistent sources, J. Phys. A, 37, 7143-7162 (2004) · Zbl 1140.35319 |
| [27] | Xiao, T.; Zeng, Y. B., A new constrained mKP hierarchy and the generalized Darboux transformation for the mKP equation with self-consistent sources, Physica A, 353, 38-60 (2005) |
| [28] | Ye, S.; Zeng, Y. B., Integration of the modified Korteweg-de Vries hierarchy with an integral type of source, J. Phys. A, 35, 283-L291 (2002) · Zbl 1039.37060 |
| [29] | Zeng, Y. B., New factorization of the Kaup-Newell hierarchy, Physica D, 73, 171-188 (1994) · Zbl 0816.35117 |
| [30] | Zeng, Y. B.; Li, Y. S., The Lax representation and Darboux transformation for constrained flows of the AKNS hierarchy., Acta Math. Sinica, New Ser., 12, 217-224 (1996) · Zbl 0867.35088 |
| [31] | Zeng, Y. B.; Ma, W. X.; Lin, R. L., Integration of the soliton hierarchy with self-consistent sources, J. Math. Phys., 41, 5453 (2000) · Zbl 0968.37023 |
| [32] | Zeng, Y. B.; Ma, W. X.; Shao, Y. J., Two binary Darboux transformations for the KdV hierarchy with self-consistent sources, J. Math. Phys., 42, 2113-2128 (2001) · Zbl 1005.37044 |
| [33] | Zeng, Y. B.; Shao, Y. J.; Ma, W. X., Integral-type Darboux transformations for the mKdV hierarchy with self-consistent sources, Commun. Theor. Phys., 38, 641-648 (2002) · Zbl 1267.37082 |
| [34] | Zeng, Y. B.; Shao, Y. J.; Xue, W. M., Negaton and positon solutions of the soliton equation with self-consistent sources, J. Phys. A, 36, 5035-5043 (2003) · Zbl 1033.35110 |
| [35] | Zhang, D. J., The \(N\)-soliton solutions for the modified KdV equation with self-consistent sources, J. Phys. Soc. Jpn., 71, 2649-2656 (2002) · Zbl 1058.37054 |
| [36] | Zhang, D. J., The \(N\)-soliton solutions of some soliton equations with self-consistent sources, Chaos Solitons Fract., 18, 31-43 (2003) · Zbl 1055.35105 |
| [37] | Zhang, D. J.; Chen, D. Y., The \(N\)-soliton solutions of the sine-Gordon equation with self-consistent sources, Phys. A, 321, 467-481 (2003) · Zbl 1011.35116 |
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