Asymptotic stability of solitons of the gKdV equations with general nonlinearity

By Yvan Martel, F. Merle. 2007

108 citations recorded by the source corpus.

AbstractWe consider the generalized Korteweg-de Vries equation (gKdV) $$\partial_t u + \partial_x (\partial_x^2 u + f(u)) = 0, \quad (t, x) \in [0, T) \times {\mathbb{R}},$$with general C3 nonlinearity f. Under an explicit condition on f and c > 0, there exists a solution in the energy space H1 of the type u(t, x) = Qc(x − x0 − ct), called soliton. In this paper, under general assumptions on f and Qc, we prove that the family of solitons around Qc is asymptotically stable in some local sense in H1, i.e. if u(t) is close to Qc (for all t ≥ 0), then u(t) locally converges in the energy space to some Qc+ as t → +∞. Note in particular that we do not assume the stability of Qc. This result is based on a rigidity property of the gKdV equation around Qc in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in Martel (SIAM J. Math. Anal. 38:759–781, 2006); Martel and Merle (J. Math. Pures Appl. 79:339–425, 2000), (Arch. Ration. Mech. Anal. 157:219–254, 2001), (Nonlinearity 1:55–80), devoted to the pure power case.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2106daa6a80b46e88cfcbb0d7a856019220b701b

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Citations in the corpus

Why are solitons stable (2008, unknown): “space or frequency, which can signiﬁcantly complicate the analysis. Remark 4.4. These arguments have recently been simpliﬁed and generalised further (to handle arbitrary power-type nonlinearities) in [52]. Interestingly, this result does not require any spectral hypotheses on the linearised problem.”
Asymptotic stability of solitons for mKdV (2015, unknown): “For perturbations in the energy space H(1), definitive asymptotic stability results in front of the solitary wave have been obtained for the whole class of subcritical gKdV equations in a series of papers by Martel-Merle [36, 37, 38].”
On the Asymptotic Stability of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}$$\end{document}-Soliton Solutions (2016, background): “For most dispersive evolution equations this is a wide open and active area of research [11,34,35,40].”
Asymptotic stability of solitons for the Benjamin-Ono equation (2008, methodology): “The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of [16], [ 18 ].”