On asymptotic dynamics for L2 critical generalized KdV equations with a saturated perturbation

Lan, Yang

doi:10.2140/apde.2019.12.43

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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

DOI: 10.2140/apde.2019.12.43

Abstract

We consider the $L^{2}$ critical gKdV equation with a saturated perturbation: $\partial_{t} u + {(u_{x x} + u^{5} - γ u | u |^{q - 1})}_{x} = 0$ , where $q > 5$ and $0 < γ ≪ 1$ . 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 $γ^{- 2 ∕ (q - 1)}$ as $γ \to 0$ ; (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 $γ = 0$ ) 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 $L^{2}$ 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.

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Keywords

gKdV, $L^2$-critical, saturated perturbation, dynamics near ground state, blow down

Mathematical Subject Classification 2010

Primary: 35Q53

Secondary: 35B20, 35B40, 37K40

Milestones

Received: 26 November 2016

Revised: 31 August 2017

Accepted: 19 April 2018

Published: 2 August 2018

Authors

Yang Lan
Laboratoire de Mathematiques D’Orsay Université Paris-Sud Orsay France

Vol. 12, No. 1, 2019