Multifunctional substitutions and soliton solutions of integrated nonlinear equations

Background. A multifunctional extension of the functional substitution method for nonlinear partial differential equations is constructed. The aim of the work is to prove the connection between the inverse problem method (IPM) and the functional substitution method (MFP), which play an important role in the modern theory of nonlinear wave processes in various types of physical systems. Such a relationship makes it possible to create an effective way to calculate solutions of integrable using the inverse problem method of equations of mathematical physics, based on functional substitutions. Materials and methods. The main method used in the work is the method of functional substitutions in the scalar and matrix of its forms. To establish the connection between the new form of solutions of the Korteweg - de Vries (KdV) type equation and the Nonlinear Schrödinger equation (NSE), the Darboux transformation method is used, which plays an important role in the IPM. Results. The paper developed a method for expanding the MFP in scalar and matrix form, which allows to obtain new integrable models of theoretical and mathematical physics along with their solutions. For the equations that are integrable with the help of the IPM, a new efficient way of constructing exact solutions equivalent to the new type of many functional substitutions is constructed using the example of the KdV and NSE equations. Conclusions. The developed approach provides a new way of constructing integrable models of theoretical and mathematical physics, along with their exact solutions.