Soliton solutions of nonlinear Schrödinger-type equations and functional substitutions

Background. The main goal of the paper is to establish the relationship between the inverse problem method (IPM) and the method of functional substitutions (MFS) in the theory of integrable nonlinear partial differential equations. The inverse problem method is used to construct solutions of equations admitting multi-soliton solutions, and the method of functional substitutions to equations, which are often called Burgers type equations. In this paper, it is demonstrated that modifying the MFS by introducing additional closing conditions into the procedure makes it possible to derive Burgers-type equations for equations that coincide with equations integrable by means of MOS. In this paper we study only equations of the type of the nonlinear Schrödinger equation (NLS), and Ginzburg - Landau equations. Materials and methods. The method of investigation is a matrix version of the method of functional substitutions. Results. In the paper, equations of Burgers type are calculated, having a form like the NLS equation for arbitrary matrix dimension of substitutions. Then, in the case of dimension n=2, all possible types of NLS-type equations are constructed. With the introduction of an additional matrix differential equation of the order of 1, equations that are identical in form to the NLS are calculated. Conclusions. The method developed in this paper establishes a connection between equations of Burgers type that are integrated with the help of the method of functional substitutions and equations integrable with the help of IPM. The above example only fixes such a connection for NLS, and in the case of matrix dimension 2, which leads to one-soliton solutions and their deformation.