  Classifications of symmetries and conservation laws are presented for a
variety of physically and analytically interesting wave equations with power
onlinearities in n spatial dimensions: a radial hyperbolic equation, a radial
Schrodinger equation and its derivative variant, and two proposed radial
generalizations of modified Korteweg--de Vries equations, as well as
Hamiltonian variants. The mains results classify all admitted local point
symmetries and all admitted local conserved densities depending on up to first
order spatial derivatives, including any that exist only for special powers or
dimensions. All such cases for which these wave equations admit, in particular,
dilational energies or conformal energies and inversion symmetries are
determined. In addition, potential systems arising from the classified
conservation laws are used to determine nonlocal symmetries and nonlocal
conserved quantities admitted by these equations. As illustrative applications,
a discussion is given of energy norms, conserved H^s norms, critical powers for
blow-up solutions, and one-dimensional optimal symmetry groups for invariant
solutions.
