Singular solutions of Clairaut-type equations in partial derivatives with reverse trigonometric functions

Background. The problem of evaluation of the special (singular) solutions of Clairaut-type partial differential equations attracts a lot of interest studying various transformations of nonlinear equations of mathematical physics, for example, Legendre transformations. Also this type of equations are significant in applied problems of theoretical physics, for example, in quantum field theory there is a connection between a special solution of the Clairaut-type equation and an effective action for composite fields. In a theory with composite fields, a one-loop effective action is determined by an equation containing an unknown nonlinear functional and its variational derivatives, which has the Clairaut-type form. The aim of this article is to find the conditions for the existence of singular solutions for the Clairaut-type partial differential equations, as well as, to obtain singular solutions for inverse trigonometric functions. The search of the special solutions of Clairaut-type partial differential equations for certain functions of derivatives in the equations remains poorly studied and is an interesting scientific field.
Materials and methods. A method is proposed for finding singular solutions of the Clairaut-type equation with a certain function on partial derivatives, using the inverse trigonometric functions as an example. The the method is to reduce the problem of finding partial derivatives of the decision function to the problem of finding convolutions of partial derivatives of the decision function with fixed parameters. The described method is applicable for finding singular solutions of the Clairaut-type equations when the function of the derivatives of the decision function has a special form.
Results. We formulate a criterion for the existence of a singular solution of the Clairaut-type partial differential equation for the case when the functions of the derivatives are the inverse trigonometric functions of linear combinations of partial derivatives. The singular solutions obtained in this paper were calculated for the case of an arbitrary number of variables, and present the main results of the paper. It is noted that in all considered cases, for a given choice of function in the equation, it is possible to solve a system that defines a singular solution of the differential equation.
Conclusions. Differential equation of the Clairaut-type is a nonlinear partial differential equations of the first order. This equation is a generalization of the well-known ordinary differential Clairaut equation. The paper describes the problem of finding a singular solution to a Clairaut-type partial differential equation for the case when the functions of the derivatives are inverse trigonometric functions. We discuss the conditions for the existence of the singular solutions and the structure of a function of derivatives for which the described method is applicable. From the course of differential equations, it is known that singular solutions of Clairaut-type partial differential equations do not always exist. Therefore, the question of finding specific functions of partial derivatives in an equation for which special solutions exist remains open and represents a promising area for further study.