A basis of approximate conservation laws for PDEs with a small parameter

The generation of new approximate conservation laws by the action of approximate Lie–Bäcklund symmetry generators on known conservation laws for a given system of perturbed or approximate partial differential equations is studied. The ideas of the classical generation theorem of conservation laws from known ones by the action of any Lie–Bäcklund symmetry generator of a given system of unperturbed partial differential equations are extended to a system of perturbed partial differential equations. We show that the generated approximate conservation laws are non-trivial if the system of the perturbed partial differential equations is derivable from a variational principle. We also determine a basis of approximate conservation laws for a given system of perturbed partial differential equations. Applications are made to perturbed linear and non-linear (1+1)(1+1) wave equations.