Analysis of nonlinear equations in private derivatives associated with the third-order scattering operator

Background. A plurality of nonlinear equations in partial derivatives having a Lax pair are either exactly integrable or equations that allow rich classes of exact solutions. The most interesting studies from a practical point of view were the development of new mathematical methods of analysis of non-linear differential equations, in particular, mathematical theory of solitons, which had huge prospects in various applications. Little research is the extensive class of nonlinear many-component equations of applied importance. The purpose of this article is to analyse this type of nonlinear equations, in particular the equation of three-wave interaction, as well as the structure of their exact solutions.
Materials and methods. Analysis of the considered nonlinear equations in the partial derivatives obtained by the Lax operator equation with first-order differential operators and third-order matrix coefficients is performed by variable substitution. This method allows them to be classified according to the main linear part and the initial equations to a simpler, equivalent species, which will be subject to further research. The traveling wave method is used to find accurate solutions.
Results. The studied nonlinear equations in the partial derivatives of the second-row with logarithmic nonlinearity belong to the Klein-Gordon class of equations and by replacing the variables their li-nee part is transformed to the hyperbolic species. Integral solutions of the studied equations in the form of traveling waves and solutions given implicitly in the form of a series have been found.
Conclusions. The results are of interest for the study of nonlinear differential equations having a Lax pair and can be used in solving applied problems of physics and engineering. These results provide an area of opportunity for studying problems of mathematical theory of solitons and can serve as a basis for further research and finding solutions to equations of this type.