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

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Abstract

We consider the L2 critical gKdV equation with a saturated perturbation: tu+(uxx+u5γu|u|q1)x=0, where q>5 and 0<γ1. For any initial data u0H1, 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 γ0; (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
Received: 26 November 2016
Revised: 31 August 2017
Accepted: 19 April 2018
First available in Project Euclid: 16 August 2018

Permanent link to this document
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

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B20: Perturbations 35B40: Asymptotic behavior of solutions 37K40: Soliton theory, asymptotic behavior of solutions

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

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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