Speaker：LAN Yang，University of Paris-Sud,Université Paris-Sud
Place：Room 1518, School of Mathematical Sciences
Detail：We consider the $L^2$ critical gKdV equation with a saturated perturbation. For any initial data in $H^1$, the corresponding solution is always global and bounded in $H^1$. This equation has a family of solitons, and our goal is to study the behavior of solutions with initial data near the soliton. Together with a suitable decay assumption, there are only 3 possibilities: i. the solution converges asymptotically to a solitary wave; ii. the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at infinite time; iii. the solution leaves any small neighborhood of the modulated family of the solitary waves. This result can be viewed as a perturbation of the rigidity dynamics near ground state for $L^2$ critical gKdV equations proved by Martel, Merle and Raphaël.
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