Duals and propagators: A canonical formalism for nonlinear equations

Abstract

A new formalism for solving general nonlinear equations is proposed. Given a nonlinear operator equation, a dual operator is constructed, canonically, similar to the adjoint operator in linear theory. By using this dual operator, advanced and retarded propagators (analogs of the Green’s functions in linear theory) are generated. These propagators satisfy the customary reciprocity and semigroup properties and yield the formal solution of the original nonlinear problem. It is further shown that these propagators can be obtained either by solving a linear equation that still contains an implicit dependence on the problem’s solution or by solving a closed-form nonlinear integral equation from which the problem’s solution itself is completely eliminated. The formalism is canonical in the sense that its applicability is not affected by the particularities of the nonlinear operator, boundary conditions, and underlying phase space. All aspects of applying this formalism to nonlinear problems are illustrated analytically on the Riccati equation.