On asymptotic dynamics for $L^2$ critical generalized KdV equations with a saturated perturbation

Abstract

We consider the $L^2$ critical gKdV equation with a saturated perturbation: ${\partial }_{t}u+{\left(u_{xx}+u^5-\gamma u|u{|}^{q-1}\right)}_x=0$, where $q>5$ and $0<\gamma \ll 1$. For any initial data $u_0\in H^1$, the corresponding solution is always global and bounded in $H^1$. This equation has a family of solutions, and our goal is to classify the dynamics near solitons. 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 ${\gamma }^{-2∕\left(q-1\right)}$ as $\gamma \to 0$; (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 classification of the rigidity dynamics near the ground state for the unperturbed $L^2$ critical gKdV (corresponding to $\gamma =0$) by Martel, Merle and Raphaël. However, the blow-down behavior (ii) is completely new, and the dynamics of the saturated equation cannot be viewed as a perturbation of the $L^2$ critical dynamics of the unperturbed equation. This is the first example of classification of the dynamics near the ground state for a saturated equation in this context. The cases of $L^2$ critical NLS and $L^2$ supercritical gKdV, where similar classification results are expected, are completely open.