Integration of the finite complex Toda lattice with a self-consistent source

In the paper, we derive a finite complex Toda lattice with a self-consistent source. We discuss the complete integrability of the constructed systems that is based on the transformation to the spectral data of an associated finite Jacobi matrix. We show that the finite complex Toda lattice with a self-consistent source is also an important theoretical model as it is a completely integrable system. Namely, we have determined the time evolution of the spectral data for the Jacobi matrix associated with the solution of a finite complex Toda chain with a self-consistent source. Then, using the solution of the inverse spectral problem with respect to the time-dependent spectral data, we recover the time-dependent Jacobi matrix and hence the desired solution of the finite complex Toda chain with a self-consistent source. For the case N=2, explicit formulas for the solution of the problem under consideration are obtained.