PIMS Applied Mathematics Seminar Series: George Bluman

  • Date: 03/23/2012
  • Time: 15:30
George Bluman

University of Saskatchewan


Nonclassical analysis of the nonlinear kompaneets equation 




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. 

Other Information: 

Location: ARTS 263


For more information please visit  University of Saskatchewan Math Department