On the asymptotic stability of $N$-soliton solutions of the three-wave resonant interaction equation

Yang, Yiling; Fan, Engui

doi:10.48550/arxiv.2101.03512

Cited by 4 publications

(5 citation statements)

References 39 publications

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“…Following the ideas in [39], it is easy to verify the L 2 -integrability of zs 21 (z) on R. In summary, we obtain the result shown in Proposition 3.2.…”

Section: The Scattering Mapssupporting

confidence: 69%

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Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data

Li,

Tian,

Yang

2021

Preprint

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In this work, we employ the ∂-steepest descent method to investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with initial conditions in weighted Sobolev space H(R). The long time asymptotic behavior of the solution q(x, t) is de-Based on the resulting asymptotic behavior, we prove the soliton resolution conjecture of the WKI equation which includes the soliton term confirmed by N(I)-soliton on discrete spectrum and the t − 1 2 order term on continuous spectrum with residual error up to O(t − 3 4 ).

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“…Following the ideas in [39], it is easy to verify the L 2 -integrability of zs 21 (z) on R. In summary, we obtain the result shown in Proposition 3.2.…”

Section: The Scattering Mapssupporting

confidence: 69%

“…Next, in order to approach our aim, it is necessary to estimate the L 2 -integral property of µ ± (z) and their derivatives. We first introduce some results of functional analysis that would be used in the following analysis [39].…”

Section: The Scattering Mapsmentioning

confidence: 99%

Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data

Li,

Tian,

Yang

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…, following the idea in [7,47], let ε 0 be an sufficiently small positive constant such that the circles {|z − z j | = ρ, |z − zj | = ρ} around z j and zj lie outside the region {z||Imz| < ε 0 }. Then, we define Ω(ξ) = 4…”

Section: The ∂ Extensions Of Jump Factorizationmentioning

confidence: 99%

On the long-time asymptotic behavior of the Camassa-Holm equation in space-time solitonic regions

Li¹,

Tian²,

Yang³

2022

Preprint

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In this work, we are devoted to study the Cauchy problem of the Camassa-Holm (CH) equation with weighted Sobolev initial data in space-time solitonic regionswhere κ is a positive constant. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the CH equation with the initial boundary value condition. Furthermore, by developing the ∂-generalization of Deift-Zhou nonlinear steepest descent method, different long-time asymptotic expansions of the solution q(x, t) are derived. Four asymptotic regions are divided in this work: For ξ ∈ −∞, − 1 4 ∪ (2, ∞), the phase function θ(z) has no stationary point on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by N (j0)-soliton on discrete spectrum with residual error up to O(t −1+2τ ); For ξ ∈ − 1 4 , 0 and ξ ∈ (0, 2), the phase function θ(z) has four and two stationary points on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by N (j0)-soliton on discrete spectrum and the t − 1 2 order term on continuous spectrum with residual error up to O(t −1 ). Our results also confirm the soliton resolution conjecture for the CH equation with weighted Sobolev initial data in space-time solitonic regions.

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“…Hereafter, the method was also developed rigorously to study the defocusing NLS under essentially minimal regularity assumptions on finite mass initial data [26] and the defocusing NLS with finite density initial data [27]. With the continuous development of the ∂ steepest descent method, more and more work has been studied, including KdV equation [28], focusing NLS equation [29], short-pluse equation [30], focusing Fokas-Lenells equation [31], modified Camassa-Holm equation [32], Wadati-Konno-Ichikawa [33] and other works [34,35,36,37].…”

Section: Introductionmentioning

confidence: 99%

The long-time asymptotic behaviors of the solutions for the coupled dispersive AB system with weighted Sobolev initial data

Wang¹,

Tian²,

Li³

2022

Preprint

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In this work, we employ the ∂-steepset descent method to study the Cauchy problem of the coupled dispersive AB system with initial conditions in weighted Sobolev space H 1,1 (R),Begin with the Lax pair of the coupled dispersive AB system, we successfully derive the solutions of the coupled dispersive AB system by constructing the basic Riemann-Hilbert problem. By using the ∂-steepset descent method, the long-time asymptotic behaviors of the solutions for the coupled dispersive AB system are characterized without discrete spectrum. Our results demonstrate that compared with the previous results, we increase the accuracy of the long-time asymptotic solution from O(t −1 log t) to O(t −1 ).

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On the asymptotic stability of $N$-soliton solutions of the three-wave resonant interaction equation

Cited by 4 publications

References 39 publications

Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data

Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data

On the long-time asymptotic behavior of the Camassa-Holm equation in space-time solitonic regions

The long-time asymptotic behaviors of the solutions for the coupled dispersive AB system with weighted Sobolev initial data

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