## PIMS Applied Mathematics Seminar Series: George Bluman

- Date: 03/23/2012
- Time: 15:30

University of Saskatchewan

Nonclassical analysis of the nonlinear kompaneets equation

Abstract:

The nonlinear Kompaneets (NLK) equation describes the spectra of photons

interacting with a rarefied electron gas. We exhibit five previously

unknown classes of explicit time-dependent solutions (each class

depending on initial conditions with two parameters) of the NLK

equation. It is shown that these solutions cannot be found as invariant

solutions using the classical Lie method (solutions obtained by

Ibragimov (2010)) but are found using the nonclassical method.

Interestingly, each of these new solutions can be expressed in terms of

elementary functions. Three of these solution classes exhibit quiescent

behaviour and the other two solution classes exhibit blow-up behaviour

infinite time. As a consequence, it is shown that corresponding

nontrivial stationary solutions are all unstable.

In the classical Lie method, one seeks symmetries that are point

transformations leaving invariant the solution manifold of a given

partial differential equation (PDE) system, i.e, symmetries that map

any solution of a given PDE system to another solution of the same

system, and then seeks corresponding solutions that are themselves

invariant. In the nonclassical method, one seeks “symmetries” that are

transformations leaving invariant a solution submanifold of a given PDE

system, i.e., “symmetries” that are transformations mapping some

solutions of a given PDE system into solutions of the same system but

map other solutions of the given PDE system map to solutions of a

different PDE system, and then seeks corresponding solutions that are

invariant. Consequently, all solutions obtainable by Lie’s classical

method can be obtained by the nonclassical method.