Nonclassical analysis of the nonlinear Kompaneets equation

The dimensional nonlinear Kompaneets (NLK) equation (Formula presented.) describes the spectra of photons interacting with a rarefied electron gas. Recently, Ibragimov obtained some time-dependent exact solutions for several approximations of this equation. In this paper, we use the nonclassical method to construct time-dependent exact solutions for the NLK equation (Formula presented.) for arbitrary constants α > 0, γ > 0. Solutions arising from “nonclassical symmetries” are shown to yield wider classes of time-dependent exact solutions for the NLK equation (Formula presented.) beyond those obtained by Ibragimov. In particular, for five classes of initial conditions, each involving two parameters, previously unknown explicit time-dependent solutions are obtained. Interestingly, each of these solutions is expressed in terms of elementary functions. Three of the classes exhibit quiescent behavior, i.e., (Formula presented.), and the other two classes exhibit blow-up behavior in finite time. As a consequence, it is shown that the corresponding nontrivial stationary solutions are unstable. © 2012, Springer Science+Business Media B.V.