The stability of solitary traveling waves in a general class of conservative nonlinear dispersive equations is discussed. A necessary condition for the exchange of stability of traveling waves is presented; an unstable eigenmode may bifurcate from the neutral translational mode only at relative extrema of the wave energy. This paper extends a result from Hamiltonian systems, and from a few integrable partial differential equations, to a broader class of conservative differential equations, with particular application to gravity-capillary surface waves.
"A stability result for solitary waves in nonlinear dispersive equations." Commun. Math. Sci. 6 (3) 791 - 797, September 2008.