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

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.