Vol. 12, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 1871–2245
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
On asymptotic dynamics for $L^2$ critical generalized KdV equations with a saturated perturbation

Yang Lan

Vol. 12 (2019), No. 1, 43–112

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.

gKdV, $L^2$-critical, saturated perturbation, dynamics near ground state, blow down
Mathematical Subject Classification 2010
Primary: 35Q53
Secondary: 35B20, 35B40, 37K40
Received: 26 November 2016
Revised: 31 August 2017
Accepted: 19 April 2018
Published: 2 August 2018
Yang Lan
Laboratoire de Mathematiques D’Orsay
Université Paris-Sud