On approximate Lagrangians and invariants for scaling reductions of a non-linear wave equation with damping

摘要

Some results on the symmetry generators of equations with a small parameter and the relationship between symmetries and conservation laws for such equations are used to construct first-integrals from the Lagrangians for similarity reductions of some classes of a non-linear wave equation. The (approximate) symmetry generators, invariants and Lagrangians maintain the perturbation order of the ‘small parameter’ stipulated in the equation – first-order in this case. These are then used to analytically find symmetry invariant solutions of the wave equation concerned.