Two-dimensional non-autonomous hyperbolic equation of the second order with power-law nonlinearities

The study of non-autonomous differential equations is of interest for the development of methods of their solutions and applications, including the study of inhomogeneous and non-stationary physical processes. In this paper, we consider the two-dimensional non-autonomous hyperbolic equation of second order containing a power-law nonlinearity in the first derivatives and arbitrary functions of the dependent and independent variables. To study the solutions of this equation, the method of functional separation of variables is used. The theorem on necessary and sufficient conditions under which this equation admits a functional separation of variables of a specified type is proved. A number of particular solutions of this equation have been obtained. In particular, we present solutions of the traveling wave type with exponential, logarithmic, and exponential functions of independent variables. For the found solutions, we formulate conditions of their existence and investigate the dependence of solutions on parameters of the equation. We have also obtained particular solutions with functions of the independent variables of a more general form. The theorem on the condition of existence of generalized self-similar solution has been proved. The results obtained in this work can be generalized for non-autonomous nonlinear equations of higher orders and more complex types of nonlinearities.

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