Наука. Инновации. Технологии (Sep 2022)
NONLINEAR EQUATION WITH THE THIRD ORDER SCATTERING OPERATOR
Abstract
Introduction: most of the differential equations associated with soliton mathematics are obtained using the Lax operator equation or the zero-curvature equation, which are the compatibility condition for a pair of linear differential systems. The case in which second-order systems were used to obtain such equations was studied in depth and comprehensively. Increasing the order of systems leads to highly overdetermined conditions. The possibility of using third-order linear systems is being studied. Materials and methods of research: methods for constructing partial differential equations using the Lax operator equation with differential operators of the first order and matrix coefficients of 3 x 3 were used. Research results: necessary and suficient conditions imposed on the parameters and func- Discussion and conclusions: tions included in the matrix coefficients, under which the commutator of two differential operators represents the multiplication operator. It is shown that the Lax equation reduces to a system of nine equations, the order of which can be reduced and reduced to one nonlinear partial differential equation. the authors demonstrated two examples of the derivation of nonlinear equations and the determination of their Lax pair. In the irst example, the main differential coeficient is considered as a lower triangular matrix, and in the second case the constant matrix has a diagonal form. As a result, second-order equations with a logarithmic nonlinearity are obtained.
