The Variational Bi-Complex for Systems of Semi-Linear Hyperbolic PDEs in Three Variables

TL;DR

This paper develops a geometric framework using the variational bi-complex to analyze semi-linear hyperbolic PDE systems in three variables, enabling the generation of conservation laws and exploring Darboux integrability.

Contribution

It extends the variational bi-complex approach to systems of semi-linear hyperbolic PDEs in three variables, introducing new methods for conservation law generation and integrability analysis.

Findings

Introduces the constrained variational bi-complex for these PDE systems.

Provides a method to generate conservation laws from adjoint solutions.

Discusses Darboux integrability and constructs infinitely many conservation laws.

Abstract

This paper extends, to a class of systems of semi-linear hyperbolic second order PDEs in three variables, the geometric study of a single nonlinear hyperbolic PDE in the plane as presented in [Anderson I.M., Kamran N., Duke Math. J. 87 (1997), 265-319]. The constrained variational bi-complex is introduced and used to define form-valued conservation laws. A method for generating conservation laws from solutions to the adjoint of the linearized system associated to a system of PDEs is given. Finally, Darboux integrability for a system of three equations is discussed and a method for generating infinitely many conservation laws for such systems is described.