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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:
 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.)
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