Orbital stability of localized structures via Bäcklund transfomations

Hoffman, A.; Wayne, C.E.

doi:10.57262/die/1360092826

Cited by 4 publications

(6 citation statements)

References 14 publications

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“…for some y(t) ∈ R. Using the Bäcklund transformation present for SG, and extensively mentioned below, Hoffman and Wayne [27] extended this stability result to the case of the kink and sketched the case of several kink structures. Inspired by this work, and using the same technique, in a recent work [60] the three main 2-soliton solutions of SG were proved to be orbitally stable for small perturbations in the energy space.…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

“…Proof of Lemma 7.1. The proof follows the ideas in [60] (see also [27] for the first approach in the SG case), with the main difference being which function will be found in terms of the others. From (7.1), (6.6) and (3.22)-(3.23), we are lead to solve the equations…”

Section: Proof Of Theorem 61: Construction Of the Manifold Of Initial...mentioning

confidence: 97%

“…The reader can consult the monograph [53] for a detailed introduction to the subject. In recent years, several works dealing with stability of solitonic structures via Bäcklund transformations have appeared: Hoffman and Wayne [27], Mizumachi and Pelinovsky [4,5,56,60], but its use as a method for proving asymptotic stability results in this paper seems to be new in nonlinear wave like equations.…”

Section: Bäcklund Transformationsmentioning

confidence: 99%

“…We lift data from a neighborhood of zero towards data near the static kink. In [27,60], such a property could not hold without the addition of an extra orthogonality condition around the kink solution (see Sect. 9 for another example).…”

Section: Lemma 36 (Parity Properties Formentioning

confidence: 99%

“…As in [4,27] and [60], the extra orthogonality condition (9.3) is essential to uniquely solve this nonlinear system. Note the similarity with (8.10).…”

Section: Lifting Of the Data In M ηmentioning

confidence: 99%

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On Asymptotic Stability of the Sine-Gordon Kink in the Energy Space

Alejo

Muñoz

Palacios

2023

Commun. Math. Phys.

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We consider the sine-Gordon (SG) equation in $$1+1$$ 1 + 1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space $$H^1\times L^2$$ H 1 × L 2 . It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space. In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in Hoffman and Wayne (Differ Int Equ 26(3–4):303–320, 2013), Muñoz and Palacios (Ann. IHP C Analyse Nonlinéaire 36(4):977–1034, 2019) and Virial estimates around odd perturbations of the vacuum solution, in the spirit of Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and $$\phi ^4$$ ϕ 4 at the level of linear Bäcklund transformations.

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Section: Introduction and Main Resultsmentioning

confidence: 96%

Section: Proof Of Theorem 61: Construction Of the Manifold Of Initial...mentioning

confidence: 97%

Section: Bäcklund Transformationsmentioning

confidence: 99%

Section: Lemma 36 (Parity Properties Formentioning

confidence: 99%

“…As in [4,27] and [60], the extra orthogonality condition (9.3) is essential to uniquely solve this nonlinear system. Note the similarity with (8.10).…”

Section: Lifting Of the Data In M ηmentioning

confidence: 99%

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On Asymptotic Stability of the Sine-Gordon Kink in the Energy Space

Alejo

Muñoz

Palacios

2023

Commun. Math. Phys.

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Stability of multi-solitons in the cubic NLS equation

Contreras

Pelinovsky

2014

J. Hyper. Differential Equations

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Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers

Alejo

Muñoz

2015

Anal. PDE

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We study the long-time dynamics of complex-valued modified Korteweg-de Vries (mKdV) solitons, which are recognized because they blowup in finite time. We establish stability properties at the H 1 level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are H 1 stable, improving our previous result [4], where we only proved H 2 stability. The main new ingredient of the proof is the use of a Bäcklund transformation which relates the behavior of breathers, complex-valued solitons and small real-valued solutions of the mKdV equation. We also prove that negative energy breathers are asymptotically stable. Since we do not use any method relying on the Inverse Scattering Transform, our proof works even under L 2 (R) perturbations, provided a corresponding local well-posedness theory is available.

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Orbital stability of localized structures via Bäcklund transfomations

Cited by 4 publications

References 14 publications

On Asymptotic Stability of the Sine-Gordon Kink in the Energy Space

On Asymptotic Stability of the Sine-Gordon Kink in the Energy Space

Stability of multi-solitons in the cubic NLS equation

Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers

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