The Coupling Algebra of Nonlinear PDE Systems: A Lie Structure on $(D, C, P)$ Triples

We introduce an algebraic structure on the space of nonlinear PDE systems. Each PDE system of dimension $n$ is encoded as a triple $\mathbf{S} = (D, C, P) \in \mathbb{R}^{n \times n} \times \mathbb{R}^{n \times n \times n} \times \mathbb{R}^{n \times n}$ of dissipation matrix, coupling tensor, and constraint projector.