Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gKdV Equations

By Yvan Martel, F. Merle, Tai-Peng Tsai. 2001

219 citations recorded by the source corpus.

We prove in this paper the stability and asymptotic stability in H1 of a decoupled sum of N solitons for the subcritical generalized KdV equations  The proof of the stability result is based on energy arguments and monotonicity of the local L2 norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [16].

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a89daacbdfe78869565d41cfc2eb7de597fe2ee1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Asymptotic stability of solitons for the Benjamin-Ono equation (2008, methodology): “The proof of Proposition 6 is exactly the same as the proof of Proposition 1 in [21], using Lemmas 10–13.”
Why are solitons stable (2008, unknown): “See [56] for an instance of this for subcritical gKdV multisolitons, and [70], [57], [69] for some results for subcritical NLS multisolitons.”
Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations (2009, methodology): “Recall that Martel, Merle and Tsai [ 21 ] proved stability and asymptotic stability of a sum of N solitons for large time for the subcritical case.”
Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations (2005, unknown): “Next, Martel, Merle and Tai-Peng Tsai [ 13 ] proved the same result in H 1 for,p = 2, 3 and a related result for p = 4. Since for p = 4 there do not exist explicit N-solitons, the results in [ 13 ] concern the stability of the sum of N solitons sufficiently decoupled in space.,family of N-soliton so”
Stability of multipeakons (2008, unknown): “Integrating by parts we compute ∫ ug2Φi = ∫ xi −∞ (u3 + uu2x − 2u2ux)Φi + ∫ +∞ xi (u3 + uu2x + 2u 2ux)Φi = Fi(u) − 4 3 u(xi) 3Φi(xi) + 2 3 ∫ xi −∞ u3Φ ′ i − 2 3 ∫ +∞ xi u3Φ ′ i .”
Asymptotic stability of solitons for mKdV (2015, unknown): “We also mention [42] on the L(2) stability of KdV solitons, [5] on the H s ≥ −1 stability of KdV solitons, [40] on N -soliton solutions of subcritical gKdV equations, and [47, 45] for a different approach.”
Multi-Soliton Solutions for the Supercritical gKdV Equations (2010, methodology): “[15] proved stability and asymptotic stability of a sum of N solitons for large time for the subcritical case.”