Instability of solitons for the critical generalized Korteweg—de Vries equation

By Yvan Martel, F. Merle. 2001

163 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/56c97ec42ce6f64677cd445e3a0228bf0aa607a2

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation (2005, background): “From [28] and Lemma 2 in [16] with Q3 replaced by the first vector of L+, one could also prove a coercivity result on L similar to Lemma 4.”
Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation (2004, methodology): “where Le = −exx + e − 5Q 4 e. As in [ 11 ], proof of Lemma 5, regularization”
Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations (2001, methodology): “Indeed, note that in (21), by integration by parts, the last term can be written as the sum of scalar products of e i , for i 3, with rapidly decaying functions (see [ 14 ], Lemma 14).”
Blow up for the critical gKdV equation III: exotic regimes (2012, background): “4) the blow up problem has been first studied in a series of works by Martel and Merle [31, 32, 44, 33, 34].”
Blow up for the critical gKdV equation II: minimal mass dynamics (2012, background): “In the series of works [26, 27, 38, 28, 29], Martel and Merle obtain the first qualitative information on the flow for small super critical mass initial data ‖Q‖L2 < ‖u0‖L2 < ‖Q‖L2 < α∗, 0 < α∗ ≪ 1, in particular the existence of finite time blow up solutions for E0 < 0 and the classification of Q a”
Blow up in finite time and dynamics of blow up solutions for the L^2-critical generalized KdV equation (2002, methodology): “Let us recall the following structural properties of L (see [7]): L(Q(3)) = −8Q(3), L(Qy) = 0, (29)”
Asymptotic stability of solitons of the gKdV equations with general nonlinearity (2007, background): “See for example [14], Proposition 1 for details.”
Asymptotic stability of solitons for the Benjamin-Ono equation (2008, background): “[4], Lemma 4.1, [ 14 ], Proposition 1 and Lemma 4). Time independent arguments.”
Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gKdV Equations (2001, unknown): “sum of N solitons”
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation (2003, background): “Let μ+ < 0 the lowest eigenvalue of L+ and φ+ a corresponding eigenvector with ‖φ+‖L2 = 1, then: Lemma 1 ([20]) There exists a universal constant δ+ > 0 such that for all ω ∈ H1 and radial, there holds (L+ω, ω) + 1 δ+ (ω, φ+)(2) ≥ δ+|ω|(2)H1 .”