This paper is concerned with studying the matrix nonlinear Schr¨odinger equation with a self-consistent source. The source consists of the combination of the eigenfunctions of the corresponding spectral problem for the matrix Zakharov-Shabat system which has not spectral singularities. The theorem about the evolution of the scattering data of a non-self-adjoint matrix Zakharov-Shabat system which potential is a solution of the matrix nonlinear Schr¨odinger equation with the self-consistent source is proved. The obtained results allow us to solve the Cauchy problem for the matrix nonlinear Schr¨odinger equation with a self-consistent source in the class of the rapidly decreasing functions via the inverse scattering method. A one-to-one correspondence between the potential of the matrix Zakharov-Shabat system and scattering data provide the uniqueness of the solution of the considering problem. A step-by-step algorithm for finding a solution to the problem under consideration is presented.

Gayrat Urazboev, Dr. Sci. (Phys.–Math.), Assoc. Prof., Urgench State University, 14, Khamid Alimdjan str., Urgench, 220100, Uzbekistan, phone: (99 891) 428-15-71, email: gayrat71@mail.ru

Anvar Reyimberganov, PhD (Phys.–Math.), Assoc. Prof., Urgench State University, 14, Khamid Alimdjan str., Urgench, 220100, Uzbekistan, phone: 9(9862) 224-67-00, email: anvar@urdu.uz

Aygul Babadjanova, PhD (Phys.–Math.), Senior Researcher, Khorezm branch of Uzbekistan Academy of Sciences V. I. Romanovskiy Institute of Mathematics, 14, Khamid Alimdjan str., Urgench, 220100, Uzbekistan, phone:(99897) 511-64-40, email: oygul@bk.ru

Urazboev G.U., Reyimberganov A.A., Babadjanova A.K. Integration of the Matrix Nonlinear Schr¨odinger Equation with a Source. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 37, pp. 63-76. https://doi.org/10.26516/1997-7670.2021.37.63

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