Abstract: 

Classical Lyapunov stability theory for solitary waves rests on a familiar scheme: modulate along the coherent-structure manifold, remove neutral directions, and prove coercivity of an augmented energy on the transverse radiation. This talk describes several settings where that scheme remains effective only after being adapted to new geometric or dynamical features. We begin with focusing NLS in a harmonic trap, where broken translation invariance is replaced by oscillating center-of-mass dynamics and a modified Lyapunov functional. We then discuss cubic–quintic NLS kinks, where nonzero boundary states require weighted metrics and a split energy argument. The second half turns to effective dynamics: first reviewing bright NLS solitons in slowly varying potentials, then describing dark-soliton dynamics for defocusing NLS, where nonzero-background variables and local virial estimates play a central role. We close briefly with stochastic KdV solitons, where Lyapunov control is combined with stochastic modulation and localized virial estimates.