Transforming Mixed Characteristic Problems into Cauchy Problems in Nonlinear Oscillation Equations

The linear formulation of a mixed characteristic problem cannot be automatically extended to the case of nonlinear equations, since the characteristics of the nonlinear equations depend on the sought solution and its lower derivatives. In this paper, an attempt is made to formulate correctly a mixed characteristic problem for the well-known nonlinear oscillation equation, which is a nonlinear analogue of the Darboux problem and consists of the simultaneous definition of a solution and its regular propagation domain. The question of solvability of the formulated problem is solved by the method of characteristics and reduction to the Cauchy problem, for what are obtained all characteristic laws of the equation and used a general integral of the equation. The proposed method to reduce a mixed characteristic problem to an initial Cauchy problem represents an approach, which can be applied to various fields requiring the analysis of nonlinear oscillations.