×
Zeng, Yunbo; Ma, Wen-Xiu; Lin, Runliang

Integration of the soliton hierarchy with self-consistent sources. (English) Zbl 0968.37023

J. Math. Phys. 41, No. 8, 5453-5489 (2000).
Summary: In contrast with the soliton equations, the evolution of the eigenfunctions in the Lax representation of soliton equation with self-consistent sources (SESCS) possesses singularity. We present a general method to treat the singularity to determine the evolution of scattering data. The AKNS hierarchy with self-consistent sources, the MKdV hierarchy with self-consistent sources, the nonlinear Schrödinger equation hierarchy with self-consistent sources, the Kaup-Newell hierarchy with self-consistent sources and the derivative nonlinear Schrödinger equation hierarchy with self-consistent sources are integrated directly by using the inverse scattering method. The \(N\) soliton solutions for some SESCS are presented. It is shown that the insertion of a source may cause the variation of the velocity of soliton. This approach can be applied to all other \((1+1)\)-dimensional soliton hierarchies.

Cited in 108 Documents

MSC:

37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51 Soliton equations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

eigenfunctions; Lax representation; soliton equation; self-consistent sources; singularity; AKNS hierarchy; MKdV hierarchy; nonlinear Schrödinger equation; Kaup-Newell hierarchy
Cite Review PDF
Full Text: DOI Link

References:

[1] Mel’nikov V. K., Commun. Math. Phys. 120 pp 451– (1989) · Zbl 0682.76013 · doi:10.1007/BF02124337
[2] Mel’nikov V. K., Commun. Math. Phys. 126 pp 201– (1989) · Zbl 0682.76013 · doi:10.1007/BF02124337
[3] DOI: 10.1103/PhysRevLett.59.2063 · doi:10.1103/PhysRevLett.59.2063
[4] DOI: 10.1088/0305-4470/23/8/013 · Zbl 0713.35084 · doi:10.1088/0305-4470/23/8/013
[5] Claud C., J. Math. Phys. 32 pp 3321– (1991) · Zbl 0744.76056 · doi:10.1063/1.529443
[6] Vlasov R. A., Dokl. Akad. Nauk BSSR 26 pp 17– (1991)
[7] DOI: 10.1080/713821151 · doi:10.1080/713821151
[8] Nakazawa M., Phys. Rev. Lett. 66 pp 2625– (1991) · doi:10.1103/PhysRevLett.66.2625
[9] Doktorov E. V., Phys. Lett. A 207 pp 153– (1995) · Zbl 1020.37525 · doi:10.1016/0375-9601(95)00678-V
[10] Shchesnovich V. S., Phys. Lett. A 213 pp 23– (1996) · Zbl 1073.35531 · doi:10.1016/0375-9601(96)00090-4
[11] Mel’nikov V. K., J. Math. Phys. 31 pp 1106– (1990) · Zbl 0705.70003 · doi:10.1063/1.528790
[12] Mel’nikov V. K., Inverse Probl. 8 pp 133– (1992) · Zbl 0752.35069 · doi:10.1088/0266-5611/8/1/009
[13] DOI: 10.1063/1.527859 · Zbl 0682.58041 · doi:10.1063/1.527859
[14] DOI: 10.1063/1.527859 · Zbl 0682.58041 · doi:10.1063/1.527859
[15] DOI: 10.1016/0375-9601(92)91052-S · doi:10.1016/0375-9601(92)91052-S
[16] DOI: 10.1088/0305-4470/26/5/018 · Zbl 0772.58019 · doi:10.1088/0305-4470/26/5/018
[17] DOI: 10.1016/0167-2789(94)90155-4 · Zbl 0816.35117 · doi:10.1016/0167-2789(94)90155-4
[18] Zeng Y., Acta Mathematica Sinica, New Series 12 pp 217– (1996) · Zbl 0867.35088 · doi:10.1007/BF02108164
[19] Mel’nikov V. K., Phys. Lett. A 133 pp 493– (1988) · doi:10.1016/0375-9601(88)90522-1
[20] Y. Zeng, Physica A 262 pp 405– (1999) · doi:10.1016/S0378-4371(98)00428-2
[21] Y. Zeng, Physica A 259 pp 278– (1998) · doi:10.1016/S0378-4371(98)00257-X
[22] Zeng Y., Phys. Lett. A 128 pp 488– (1991)
[23] DOI: 10.1063/1.523737 · Zbl 0383.35015 · doi:10.1063/1.523737
[24] Zhuang D., Beijing Daxue Xuebao 3 pp 1– (1981)
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.