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We construct invariant manifolds of interior multi-spike states for the nonlinear Cahn-Hilliard equation and then investigate the dynamics on it. An equation for the motion of the spikes is derived. It turns out that the dynamics of interior spikes has a global character and each spike interacts with all the others and with the boundary. Moreover, we show that the speed of the interior spikes is super slow, which indicates the long time existence of dynamical multi-spike solutions in both positive and negative time. This result is the application of an abstract result concerning the existence of truly invariant manifolds with boundary when one has only approximately invariant manifolds. The abstract result is an extension of one by P. Bates, K. Lu, and C. Zeng to the case of a manifold with boundary, consisting of almost stationary states.

This work is devoted to study the dynamics of the supercritical gKDV equations near solitary waves in the energy space H 1 . We construct smooth local center-stable, center-unstable and center manifolds near the manifold of solitary waves and give a detailed description of the local dynamics near solitary waves. In particular, the instability is characterized as follows: any forward flow not starting from the center-stable manifold will leave a neighborhood of the manifold of solitary waves exponentially fast. Moreover, orbital stability is proved on the center manifold, which implies the uniqueness of the center manifold and the solutions on it exist globally and asymptotically approach the solitary waves.

We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (for example the Whitham equation, the generalized KDV equation, the Benjamin–Ono equation), the nonlinear Schrodinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs. Second, for the KDV type equations the loss of derivative in the nonlinear terms is overcome in two complementary cases: (1) for smooth nonlinear terms and general dispersive operators, we construct higher order approximation solutions and then use energy type estimates; (2) for nonlinear terms of low regularity, with some additional assumptions on the dispersive operator, we use a bootstrap argument to overcome the loss of a derivative.

We study the local dynamics near general unstable traveling waves of the 3D Gross–Pitaevskii equation in the energy space by constructing smooth local invariant center-stable, center-unstable and center manifolds. We also prove that (i) the center unstable manifold attracts nearby orbits exponentially before they go away from the traveling waves along the center or unstable directions and (ii) if an initial data is not on the center-stable manifolds, then the forward orbit leaves traveling waves exponentially fast. Furthermore, under an additional non-degeneracy assumption, we show the orbital stability of the travelingwaves on the centermanifolds,which also implies the uniqueness of the local invariant manifolds. Our method based on a geometric bundle coordinate system should work for a general class of Hamiltonian PDEs.