Conservation Laws and Symmetries of Semilinear Radial Wave Equations

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.Comment: 16 pages. Final version with minor revision

Potential equivalence transformations for nonlinear diffusion-convection equations

Potential equivalence transformations (PETs) are effectively applied to a class of nonlinear diffusion-convection equations. For this class all possible potential symmetries are classified and a theorem on connection of them with point ones via PETs is also proved. It is shown that the known non-local transformations between equations under consideration are nothing but PETs. Action of PETs on sets of exact solutions of a fast diffusion equation is investigated.Comment: 10 page

Potential Nonclassical Symmetries and Solutions of Fast Diffusion Equation

The fast diffusion equation $u_t=(u^{-1}u_x)_x$ is investigated from the symmetry point of view in development of the paper by Gandarias [Phys. Lett. A 286 (2001) 153-160]. After studying equivalence of nonclassical symmetries with respect to a transformation group, we completely classify the nonclassical symmetries of the corresponding potential equation. As a result, new wide classes of potential nonclassical symmetries of the fast diffusion equation are obtained. The set of known exact non-Lie solutions are supplemented with the similar ones. It is shown that all known non-Lie solutions of the fast diffusion equation are exhausted by ones which can be constructed in a regular way with the above potential nonclassical symmetries. Connection between classes of nonclassical and potential nonclassical symmetries of the fast diffusion equation is found.Comment: 13 pages, section 3 is essentially revise

Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations

A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations $f(x)u_{tt}=(H(u)u_x)_x+K(u)u_x$, is given, by using a compatibility method and additional equivalence transformations. A number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Furthermore, the possible additional equivalence transformations between equations from the class under consideration are investigated. Exact solutions of special forms of these equations are also constructed via classical Lie method and generalized conditional transformations. Local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations.Comment: 23 page