## Analysis & PDE

• Anal. PDE
• Volume 12, Number 1 (2019), 43-112.

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

Yang Lan

#### Abstract

We consider the $L2$ critical gKdV equation with a saturated perturbation: $∂tu+(uxx+u5−γu|u|q−1)x=0$, where $q>5$ and $0<γ≪1$. For any initial data $u0∈H1$, the corresponding solution is always global and bounded in $H1$. 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 $H1$ norm is of size $γ−2∕(q−1)$ as $γ→0$; (ii) the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at $+∞$; (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 $L2$ critical gKdV (corresponding to $γ=0$) 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 $L2$ 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 $L2$ critical NLS and $L2$ supercritical gKdV, where similar classification results are expected, are completely open.

#### Article information

Source
Anal. PDE, Volume 12, Number 1 (2019), 43-112.

Dates
Revised: 31 August 2017
Accepted: 19 April 2018
First available in Project Euclid: 16 August 2018

https://projecteuclid.org/euclid.apde/1534384914

Digital Object Identifier
doi:10.2140/apde.2019.12.43

Mathematical Reviews number (MathSciNet)
MR3842909

Zentralblatt MATH identifier
06930184

#### Citation

Lan, Yang. On asymptotic dynamics for $L^2$ critical generalized KdV equations with a saturated perturbation. Anal. PDE 12 (2019), no. 1, 43--112. doi:10.2140/apde.2019.12.43. https://projecteuclid.org/euclid.apde/1534384914

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