Title:  On Higher Order Stability Theory of Solitary Waves.

Abstract:  Consideration is given to the stability of solitary-wave
solutions of several of the most prominent soliton equations in the
Sobolev spaces H^n, for all n=1,2,3.....In particular, the stability in
higher-order spaces means practically that not only does the bulk of what
emanates from the perturbed solitary wave stay close in shape and
propagation speed to the original solitary wave, but emerging residual
oscillations must also be very small and not only in the energy norm.
The talk is organized into two parts.  The first part employs conserved
integrals involving n^th-derivatives and the stability results already
established in the lower-order Sobolev spaces to show that solitary-wave
solutions are stable in H^n, for arbitrarily large n.  The theory
therefore applies to the completely integrable Hamiltonian equations such
as the KdV, mKdV, Benjamin-Ono, Intermediate Long Wave and Nonlinear Cubic
Schrodinger equations.  The concentration compactness method is used in
the other part to demonstrate that the solitary-wave solutions of the KdV
equation are stable in H^n.