On Laplace Invariants of a Hyperbolic Equation with Mixed Derivative and Quadratic Nonlinearities

Under study is some two-dimensional second order nonlinear hyperbolicequation with variable coefficients. The left-hand side of the equation contains quadraticnonlinearities with the unknown function and its derivatives. We consider the linear multiplicativetransformations of the unknown function which preserve the form of the initial equation. By analogy with linear equations,the Laplace invariants are determined as the invariants of such transformation.We find the expressions of the Laplaceinvariants which use the coefficients of the equation and their first derivatives.Also, we consider the general case and the case that some coefficients of the equation equal to zeroas well as prove the main theorem aboutLaplace invariants.By the theorem, two nonlinear hyperbolic equations under studycan be connected by using a linear multiplicative transformation if only if the Laplace invariantsfor both equations are the same. We find the equivalent systems of the first order equationscontaining the Laplace invariants, in the general case and in the case that somecoefficients of the equation vanish. We demonstrate that a solution to the initial equation can bereceived in quadratures under some additional conditions on the coefficients and the Laplace invariants.