| Consideration is given to the stability of solitary-wave solutions of several of the most prominent soliton equations in the Sobolev spaces Hn(R), 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 manuscript 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 Hn, for arbitrary 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 H2. A conjecture is made for the Hn-stability for n arbitrarily large and a procedure on how to handle this case is suggested. The stability of n-soliton solutions is also investigated. |
