On Laplace invariants of two-dimensional hyperbolic equation with mixed derivative and quadratic nonlinearities

We study two-dimensional nonlinear hyperbolic equation of the second order with variable coefficients. The left side of this equation contains quadratic nonlinearities on unknown function and its derivatives. We consider a set of linear multiplicative transformations of unknown function which keep a form of initial equation. By analogy with linear equations, the Laplace invariants are determined as the invariants of this transformation. Expressions for the Laplace invariants over the coefficients of the equation and their first derivatives are obtained. We consider both the general case and the case when some coefficients of the equation equals to zero. The main theorem about Laplace invariants is proved. According to this theorem, two nonlinear hyperbolic equations of the considering form can be connected with the help of linear multiplicative transformation if only if the Laplace invariants for both equations have the same values. We have found the equivalent systems of the first order equations, containing the Laplace invariants, for considering equation in general case and in the case when some coefficients of the equation equals to zero. It is shown that the solution of the initial equation can be received in quadratures if some additional conditions on the coefficients and on the Laplace invariants are fulfilled.