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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 establish the existence of a global invariant manifold of bubble states for the mass-conserving Allen-Cahn Equation in two space dimensions and give the dynamics for the center of the bubble.