### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 096, 49 pages      arXiv:1712.03068      https://doi.org/10.3842/SIGMA.2018.096

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

Sara Froehlich
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC H3A 0B9 Canada

Received December 11, 2017, in final form August 24, 2018; Published online September 09, 2018

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.

Key words: Laplace transform; conservation laws; Darboux integrable; variational bi-complex; hyperbolic second-order equations.

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