#### Vol. 12, No. 1, 2019

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

### Yang Lan

Vol. 12 (2019), No. 1, 43–112
##### 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.

##### Keywords
gKdV, $L^2$-critical, saturated perturbation, dynamics near ground state, blow down
##### Mathematical Subject Classification 2010
Primary: 35Q53
Secondary: 35B20, 35B40, 37K40