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Zbl 0968.37023
Zeng, Yunbo; Ma, Wen-Xiu; Lin, Runliang
Integration of the soliton hierarchy with self-consistent sources. (English)
[J] J. Math. Phys. 41, No.8, 5453-5489 (2000). ISSN 0022-2488; ISSN 1089-7658/e

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.

MSC 2000:: *37K15 Integration by inverse spectral and scattering methods
35Q51 Solitons
37K10 Completely integrable systems etc.

Keywords: eigenfunctions; Lax representation; soliton equation; self-consistent sources; singularity; AKNS hierarchy; MKdV hierarchy; nonlinear Schrödinger equation; Kaup-Newell hierarchy

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Zentralblatt MATH Berlin [Germany]