Lan Yang: On asymptotic dynamics for $L^2$-critical gKdV with saturated perturbations

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Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	Lan Yang, Universitat Basel
Inviter:
Title:	On asymptotic dynamics for $L^2$-critical gKdV with saturated perturbations
Time & Venue:	2019.7.9 10:45-11:45 N224
Abstract:	We consider the $L^2$ critical gKdV equation with a saturated perturbation. In this case, all $H^1$ solution are global in time. Our goal is to classify the asymptotic dynamics for solutions with initial data near the ground state. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave, whose $H^1$ norm is of size $\gamma^{-2/(q-1)}$, as $\gamma\rightarrow0$; (ii) the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at $+\infty$; (iii) the solution leaves any small neighborhood of the modulated family of the solitary waves. This extends the result of classification of the rigidity dynamics near the ground state for the unperturbed $L^2$ critical gKdV (corresponding to $\gamma=0$) by Martel, Merle and Rapha\"el. It also provides a way to consider the continuation properties after blow-up time for $L^2$-crtitical gKdV equations.