Abstract:
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 on unknown function and its derivatives. We consider a set of linear multiplicative transformations of the 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 are obtained through the coefficients of the equation and their first derivatives. For the equations under consideration, equivalent systems of first-order equations are found, containing the Laplace invariants. It is shown that if one of the Laplace invariants is equal to zero, then the corresponding system is reduced to a single first-order equation. Also in this case, if certain additional conditions on the coefficients are met, a solution to the original equation in quadratures can be obtained. The studies were carried out for a hyperbolic equation with a mixed derivative and for a nonlinear second-order equation of general form with a homogeneous polynomial of the second degree in the unknown function and its derivatives. In these cases, expressions for the Laplace invariants are obtained and the corresponding equivalent systems are given.
Keywords:partial differential equation, hyperbolic equation, Laplace invariant, linear multiplicative transformation
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