Nonlinear stability of 2-solitons of the sine-Gordon equation in the energy space

Muñoz, Claudio; Palacios, José M.

doi:10.1016/j.anihpc.2018.10.005

Cited by 25 publications

(73 citation statements)

References 34 publications

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“…The proof of Theorem 2.1 is simple, variational and follows previous ideas presented in [4] for the case of mKdV breathers, and [6] for the case of the Sine-Gordon breather (see also [33] for a recent improvement of this last result, based in [5]). The main differences are in the complex-valued nature of the involved breathers, and the nonlocal character of the KM and P breathers.…”

Section: Resultsmentioning

confidence: 62%

Stability and instability of breathers in the $U(1)$ Sasa-Satusuma and Nonlinear Schrödinger models

Alejo¹,

Fanelli²,

Muñoz³

2019

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We consider the Sasa-Satsuma (SS) and Nonlinear Schrödinger (NLS) equations posed along the line, in 1+1 dimensions. Both equations are canonical integrable U (1) models, with solitons, multi-solitons and breather solutions [43]. For these two equations, we recognize four distinct localized breather modes: the Sasa-Satsuma for SS, and for NLS the Satsuma-Yajima, Kuznetsov-Ma and Peregrine breathers. Very little is known about the stability of these solutions, mainly because of their complex structure, which does not fit into the classical soliton behavior [17]. In this paper we find the natural H 2 variational characterization for each of them, and prove that Sasa-Satsuma breathers are H 2 nonlinearly stable, improving the linear stability property previously proved by Pelinovsky and Yang [36]. Moreover, in the SS case, we provide an alternative understanding of the SS solution as a breather, and not only as an embedded soliton. The method of proof is based in the use of a H 2 based Lyapunov functional, in the spirit of [4], extended this time to the vectorvalued case. We also provide another rigorous justification of the instability of the remaining three nonlinear modes (Satsuma-Yajima, Peregrine y Kuznetsov-Ma), based in the study of their corresponding linear variational structure (as critical points of a suitable Lyapunov functional), and complementing the instability results recently proved e.g. in [32].

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Section: Resultsmentioning

confidence: 62%

Stability and instability of breathers in the $U(1)$ Sasa-Satusuma and Nonlinear Schrödinger models

Alejo¹,

Fanelli²,

Muñoz³

2019

Preprint

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“…Theorem 5.1 is in contrast with other positive results involving breather solutions [36,38,47]. In those cases, the involved equations (mKdV, Sine-Gordon) were globally well-posed in the energy space (and even in smaller subspaces), with uniform in time bounds.…”

Section: The Peregrine Breathermentioning

confidence: 80%

“…In this paper, we review the known results about stability in Sobolev spaces of the Peregrine (P) breather 1 [1]:…”

Section: Introductionmentioning

confidence: 99%

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Review on the Stability of the Peregrine and Related Breathers

2020

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In this note, we review stability properties in energy spaces of three important nonlinear Schrödinger breathers: Peregrine, Kuznetsov-Ma, and Akhmediev. More precisely, we show that these breathers are unstable according to a standard definition of stability. Suitable Lyapunov functionals are described, as well as their underlying spectral properties. As an immediate consequence of the first variation of these functionals, we also present the corresponding nonlinear ODEs fulfilled by these nonlinear Schrödinger breathers. The notion of global stability for each breather mentioned above is finally discussed. Some open questions are also briefly mentioned.

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“…We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in [24,52] and Virial estimates around odd perturbations of the vacuum solution, in the spirit of [32], 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.…”

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confidence: 99%

On the asymptotic stability of the sine-Gordon kink in the energy space

Alejo¹,

Muñoz²,

Palacios³

2020

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We consider the sine-Gordon (SG) equation in 1+1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space 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 [24,52] and Virial estimates around odd perturbations of the vacuum solution, in the spirit of [32], 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 [32], 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 φ 4 at the level of linear Bäcklund transformations. Date: March 23, 2020. * M.A. Alejo was partially supported by CNPq grant no. 305205/2016-1. * * C. M. work was funded in part by Chilean research grants FONDECYT 1191412, project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001. * * * J. M. P. was partially supported by Chilean research grants FONDECYT 1191412 and project France-Chile ECOS-Sud C18E06.

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Nonlinear stability of 2-solitons of the sine-Gordon equation in the energy space

Cited by 25 publications

References 34 publications

Stability and instability of breathers in the $U(1)$ Sasa-Satusuma and Nonlinear Schrödinger models

Stability and instability of breathers in the $U(1)$ Sasa-Satusuma and Nonlinear Schrödinger models

Review on the Stability of the Peregrine and Related Breathers

On the asymptotic stability of the sine-Gordon kink in the energy space

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