This article is cited in 2 papers

Integro-differential equations and functional analysis

Integration of the matrix nonlinear Schrödinger equation with a source

G. U. Urazboev^a, A. A. Reyimberganov^a, A. K. Babadjanova<sup>b

a</sup> Urgench State University, Urgench, Uzbekistan
^b V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Urgench, Uzbekistan

Abstract: This paper is concerned with studying the matrix nonlinear Schrödinger 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 Schrödinger equation with the self-consistent source is proved. The obtained results allow us to solve the Cauchy problem for the matrix nonlinear Schrödinger 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.

Keywords: matrix nonlinear Schrödinger equation, self-consistent source, inverse scattering method, scattering data.

UDC: 517.957

MSC: 34L25, 35Q41, 35R30, 34M46

Received: 08.05.2021

Language: English

DOI: 10.26516/1997-7670.2021.37.63

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