Multiplication for solutions of the equation $\grad{f} = M\grad{g}$

TL;DR

This paper introduces a bilinear multiplication on solutions of certain first-order PDE systems, enabling algebraic construction of solutions and revealing that all solutions can be expressed as power series of simple solutions.

Contribution

It defines a novel bilinear $*$-multiplication on solution spaces of gradient PDEs, generalizing superposition principles and providing a new way to represent solutions as power series.

Findings

The $*$-multiplication acts as a nonlinear superposition principle.

All solutions can be expressed as power series of simple solutions.

The approach applies to a broad class of gradient PDE systems.

Abstract

Linear first order systems of partial differential equations of the form $\nabla f = M\nabla g,$ where $M$ is a constant matrix, are studied on vector spaces over the fields of real and complex numbers, respectively. The Cauchy--Riemann equations belong to this class. We introduce a bilinear $*$-multiplication on the solution space, which plays the role of a nonlinear superposition principle, that allows for algebraic construction of new solutions from known solutions. The gradient equations $\nabla f = M\nabla g$ constitute only a simple special case of a much larger class of systems of partial differential equations which admit a bilinear multiplication on the solution space, but we prove that any gradient equation has the exceptional property that the general analytic solution can be expressed through power series of certain simple solutions, with respect to the $*$-multiplication.