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Integration of the soliton hierarchy with self-consistent sources. (English) Zbl 0968.37023
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:
37K15Integration of completely integrable systems by inverse spectral and scattering methods
35Q51Soliton-like equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies