On some new solutions of the multi-dimensional first order partial differential equation with power-law non-linearities

Investigations of nonlinear partial differential equations of the first order with an arbitrary number of independent variables are an important part of up-to-date mathematical physics. For many equations of this class, only solutions of the simplest kind are known, in particular, solutions of the travelling wave type. The present work is devoted to finding solutions of a more complex form for the multi-dimensional equation of the first order with power-law non-linearity in derivatives.
To solve this problem, in this paper we propose a new variant of the method of separation of variables — the method of two-level functional separation of variables. The characteristic feature of this method is that the desired function depends on a superposition of functions of the first and second levels of one variable, and these functions are determined as the result of solving some ordinary differential equations.
Based on the method proposed in the paper, new exact solutions of the considered equation are obtained in an implicit form. The solutions contain some generalized polynomials of independent variables. Conditions of the existence of these solutions are specified. The results of this work can be generalized to other non-linear first order equations and equations of higher orders with many independent variables.