Stability in H1/2 of the sum of K solitons for the Benjamin–Ono equation

Gustafson, Stephen J.; Takaoka, Hideo; Tsai, Tai‐Peng

doi:10.1063/1.3032578

Cited by 11 publications

(12 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Integrating (12) with respect t, we obtain (iii). To prove (iv), we first write u 0 = u 0,1 + u 0,2 whereû 0,1 (ξ) = χ |ξ|≤Mû0 (ξ), for M > 0 to be chosen.…”

Section: Plan Of the Papermentioning

confidence: 99%

Local well-posedness and blow-up in the energy space for a class of $ L^{2} $ critical dispersion generalized Benjamin–Ono equations

Kenig

Martel

Robbiano

2011

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

We consider a family of dispersion generalized Benjamin-Ono equations (dgBO)where |D| α u = |ξ| α u and 1 ≤ α ≤ 2. These equations are critical with respect to the L 2 norm and global existence and interpolate between the modified BO equation (α = 1) and the critical gKdV equation (α = 2).First, we prove local well-posedness in the energy space for 1 < α < 2, extending results in [19]-[20] for the generalized KdV equations.Second, we address the blow up problem in the spirit of [25,30] concerning the critical gKdV equation, by studying rigidity properties of the (dgBO) flow in a neighborhood of the solitons. We prove that for α close to 2, solutions of negative energy close to solitons blow up in finite or infinite time in the energy space H Ensuite, nousétudions le phénomène d'explosion dans l'esprit de [25,30] concernant l'équation de gKdV critique, enétudiant les propriétés de rigidité du flot de (dgBO) dans un voisinage des solitons. Nous montrons que pour α proche de 2, les solutions d'énergie négative proches des solitons explosent en temps fini ou infini dans l'espace d'énergie H α 2 . La preuve de ce résultat d'explosion nécessite d'une part l'adaptationà (dgBO) de résultats de monotonie de normes L 2 locales par des méthodes d'opérateurs pseudodifferentiels et d'autre part des arguments de perturbation proche du cas (gKdV) pour obtenir des propriétés structurelles du flot linéarisé autour des solitons.2 Local well-posedness in the energy space Proof of Theorem 1We denote the Fourier transform by F(f )(ξ) =f (ξ) = e −ixξ f (x)dx.We introduce the group W α (t) defined by F(W α (t)f )(ξ) = e it(|ξ| α ξ) f (ξ), 1 < α < 2.Then, we claim (or recall) the following linear estimates (we use classical notation from [20]).

show abstract

“…Integrating (12) with respect t, we obtain (iii). To prove (iv), we first write u 0 = u 0,1 + u 0,2 whereû 0,1 (ξ) = χ |ξ|≤Mû0 (ξ), for M > 0 to be chosen.…”

Section: Plan Of the Papermentioning

confidence: 99%

Local well-posedness and blow-up in the energy space for a class of $ L^{2} $ critical dispersion generalized Benjamin–Ono equations

Kenig

Martel

Robbiano

2011

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

show abstract

“…After the paper was finished and submitted, we learned that S. Gustafson, H. Takaoka, and T-P. Tsai [10] have obtained independently the stability part of Theorem 4. Note that the main result of the present paper, i.e.…”

Section: Stability Of Soliton In the Energy Space ([3] [31]) There mentioning

confidence: 99%

“…Note that the main result of the present paper, i.e. asymptotic stability of (single or multi-) solitons is not addressed in [10].…”

Section: Stability Of Soliton In the Energy Space ([3] [31]) There mentioning

confidence: 99%

Asymptotic stability of solitons for the Benjamin-Ono equation

Kenig¹,

Martel²

2009

Rev. Mat. Iberoam.

View full text Add to dashboard Cite

show abstract

“…Of course, in the case α = 1, the N -soliton solutions of the Benjamin-Ono equation are explicit by using inverse scattering method [40], [16], [8], [59], [51]. These N -solitons were also proved to be orbitally stable by Neves and Lopes [60] and Gustafson, Takaoka and Tsai [23] and even asymptotically stable by Kenig and Martel [34].…”

Section: Solitary Wave Solutionsmentioning

confidence: 99%

Asymptotic $N$-soliton-like solutions of the fractional Korteweg-de Vries equation

Eychenne¹

2021

Preprint

View full text Add to dashboard Cite

We construct N -soliton solutions for the fractional Korteweg-de Vries (fKdV) equationin the whole sub-critical range α ∈] 1 2 , 2[. More precisely, if Qc denotes the ground state solution associated to fKdV evolving with velocity c, then given 0where ρ ′ j (t) ∼ c j as t → +∞. The proof adapts the construction of Martel in the generalized KdV setting [Amer. J. Math. 127 (2005), pp. 1103-1140]) to the fractional case. The main new difficulties are the polynomial decay of the ground state Qc and the use of local techniques (monotonicity properties for a portion of the mass and the energy) for a non-local equation. To bypass these difficulties, we use symmetric and non-symmetric weighted commutator estimates. The symmetric ones were proved by Kenig, Martel and Robbiano [Annales de l'IHP Analyse Non Linéaire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new.

show abstract

Stability in H1/2 of the sum of K solitons for the Benjamin–Ono equation

Abstract: This note proves the orbital stability in the energy space H 1/2 of the sum of widelyspaced 1-solitons for the Benjamin-Ono equation, with speeds arranged so as to avoid collisions.

Cited by 11 publications

References 21 publications

Local well-posedness and blow-up in the energy space for a class of \( L^{2} \) critical dispersion generalized Benjamin–Ono equations

Local well-posedness and blow-up in the energy space for a class of \( L^{2} \) critical dispersion generalized Benjamin–Ono equations

Asymptotic stability of solitons for the Benjamin-Ono equation

Asymptotic $N$-soliton-like solutions of the fractional Korteweg-de Vries equation

Contact Info

Product

Resources

About