Implicit Solutions of PDE's

TL;DR

This paper explores implicit solutions to Lorentz-invariant nonlinear PDEs, focusing on second-order equations solved by quadratic relationships among variables, and discusses their relation to general solutions.

Contribution

It characterizes second-order PDEs solvable by inhomogeneous quadratic relationships with variable coefficients, especially when the discriminant vanishes, leading to implicit solutions of the Universal Field Equation.

Findings

Implicit solutions obtained when the quadratic discriminant vanishes.

Conditions for second-order PDEs solvable by quadratic relationships.

Relation between implicit solutions and the general solution of the Universal Field Equation.

Abstract

Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a single unknown $\phi$ are solved by the imposition of an inhomogeneous quadratic relationship among the independent variables, whose coefficients are functions of $\phi$ is discussed, and it is shown that if the discriminant of the quadratic vanishes, then an implicit solution of the so-called Universal Field Equation is obtained. The relation to the general solution is discussed.