Vol. 12, No. 1, 2019

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 L2 critical gKdV equation with a saturated perturbation: tu + (uxx + u5 γu|u|q1)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 γ2(q1) 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.

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