PIMS Applied Mathematics Seminar Series: George Bluman

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.