On Laplace Invariants of Two-Dimensional Nonlinear Equations of the Second Order with Homogeneous Polynomial

We study two-dimensional nonlinear partial differential equations of the second order with variable coefficients. The left-hand side of these equations is a homogeneous polynomial of the second degree in unknown function and its derivatives. We consider a set of linear multiplicative transformations of the unknown function which keep the form of the initial equation. By analogy with linear equations, the Laplace invariants are determined as the invariants of this transformation. The expressions for the Laplace invariants in terms of the coefficients of the equation and their first derivatives are obtained. For the considered equations, the equivalent systems of the first order equations containing the Laplace invariants are found. It is shown that if one of the Laplace invariants equals zero, the corresponding system is reduced to one first-order equation. In addition, in this case the solution to the initial equation can be obtained in quadratures if some additional conditions on the coefficients are met. The investigations are executed for a hyperbolic equation with a mixed derivative and for a nonlinear second-order equation of the general form with a homogeneous second-degree polynomial in the unknown function and its derivatives. For these cases we obtained the Laplace invariants and the equivalent systems of first-order equations.