题目：Classification of dynamics near the solitary waves of the supercritical gKDV equations.
摘要：We classify the local dynamics near solitary waves of the supercritical gKDV equations. We proved that there exists a co-dimensional one center-stable (unstable) manifold, such that if the initial data is not on the center-stable (un-stable) manifold then the corresponding forward(backward) flow will be ejected away from the solitons exponentially fast; There exists a co-dimensional two center manifold, such that if the initial data is not on the center manifold, then the flow will get away from the solitons exponentially fast either in positive time or in negative time. Moreover, we show the orbital stability of the solitons on the center manifold, which also implies the global existence of the solutions on the center manifold and the local uniqueness of the center manifolds. Further-more, applying a theorem of Martel and Merle, we have that the solitons are asymptotically stable on the center manifold in some local sense. This is a joint work with Zhiwu Lin and Chongchun Zeng.