A basis of conservation laws for partial differential equations

The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical Lie-Backlund generator is extended to include any Lie-Backlund generator. Also, it is shown that the Lie algebra of Lie-Backlund symmetries of a conserved vector of a system is a subalgebra of the Lie-Backlund symmetries of the system. Moreover, we investigate a basis of conservation laws for a system and show that a generated conservation law via the action of a symmetry operator which satisfies a commutation rule is nontrivial if the system is derivable from a variational principle. We obtain the conservation laws of a class of nonlinear diffusion-convection and wave equations in (1+1)-dimensions. In fact we find a basis of conservation laws for the diffusion equations in the special case when it admits proper Lie-Backlund symmetries. Other examples are presented to illustrate the theory.