Lax representation with first-order operators for new nonlinear Korteweg - de Vries type equations

Background. In this work, a new representation is constructed for equations of the Korteweg - de Vries (KdV) type. The proposed approach allows to obtain a universal Lax representation for a set of nonlinear partial differential equations, for which such a representation was not previously known. Materials and methods. The construction of the Lax representation for the new equations is based on the reduction of the general compatibility condition for two nonlinear first-order equations with a polynomial dependence on the unknown function. Results. A new general scheme for calculating the Lax representations in the form of two linear operators of the first order with a spectral parameter for the set of 1 + 1 equations integrable using the inverse problem method is obtained in this work. Infinite series of differential conservation laws for these equations are calculated and a special type of Backlund transformations for them is indicated. Conclusions. For a whole class of equations of the KdV-type, there is a general form of Lax representations that allows the inverse problem method to be applied to them.