Nonlinear Stability of MKdV Breathers

Alejo, Miguel A.; Muñoz, Claudio

doi:10.1007/s00220-013-1792-0

Cited by 56 publications

(126 citation statements)

References 46 publications

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“…It is well-known that (1.1) may have soliton (or solitary wave) solutions of the form u(t, x) = Q c (x − ct), c > 0, (1.5) where Q c is a stationary solution to the ODE Q ′′ c − cQ c + f (Q c ) = 0, Q c ∈ H 1 (R), (1.6) provided f satisfies standard assumptions. Additionally, it is well-known that both mKdV and Gardner models do have stable breather solutions [1,3,4,6,2], that is to say, localized in space solutions which are also periodic in time, up to the symmetries of the equation. An example of these type of solutions is the mKdV breather: for any α, β > 0,…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…β = ± √ 3α. From [3], it is well-known that small H 1 breathers are characterized by the constraints β small ("small mass") and α 2 β also small ("small energy"). These two conditions are clearly not in contradiction with the additional assumption γ = 0.…”

Section: )mentioning

confidence: 99%

“…For further results on the mKdV equation under initial conditions in weighted spaces, see [15,16,13]. See also [3,4,6] for more properties on mKdV breathers. Remark 1.15.…”

Section: )mentioning

confidence: 99%

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Breathers and the Dynamics of Solutions in KdV Type Equations

Muñoz

Ponce

2018

Commun. Math. Phys.

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In this paper our first aim is to identify a large class of non-linear functions f (·) for which the IVP for the generalized Korteweg-de Vries equation does not have breathers or "small" breathers solutions. Also we prove that all small, uniformly in time L 1 ∩ H 1 bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order t 1/2 around any compact set in space. This set is included in the linearly dominated dispersive region x ≪ t. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

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Breathers and the Dynamics of Solutions in KdV Type Equations

Muñoz

Ponce

2018

Commun. Math. Phys.

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“…Important examples of breather solutions are the modified KdV [79,55] and the Sine-Gordon [55] breathers. For recent works in the subject of stability, see [3,4,71,72,2,6,7]. CH peakons and BBM solitons are not zero-speed solutions.…”

Section: Solitons and Peakonsmentioning

confidence: 99%

On the Dynamics of Zero-Speed Solutions for Camassa–Holm-Type Equations

Alejo

Cortez

Kwak

et al. 2019

International Mathematics Research Notices

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In this paper we consider globally defined solutions of Camassa-Holm (CH) type equations outside the well-known nonzero speed, peakon region. These equations include the standard CH and Degasperis-Procesi (DP) equations, as well as nonintegrable generalizations such as the b-family, elastic rod and BBM equations. Having globally defined solutions for these models, we introduce the notion of zero-speed and breather solutions, i.e., solutions that do not decay to zero as t → +∞ on compact intervals of space. We prove that, under suitable decay assumptions, such solutions do not exist because the identically zero solution is the global attractor of the dynamics, at least in a spatial interval of size |x| t 1/2− as t → +∞. As a consequence, we also show scattering and decay in CH type equations with long range nonlinearities. Our proof relies in the introduction of suitable Virial functionalsà la Martel-Merle in the spirit of the works [74,75] and [50] adapted to CH, DP and BBM type dynamics, one of them placed in L 1x , and a second one in the energy space H 1 x . Both functionals combined lead to local in space decay to zero in |x| t 1/2− as t → +∞. Our methods do not rely on the integrable character of the equation, applying to other nonintegrable families of CH type equations as well.

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“…See also [8,9] for numerical studies of the stability of mKdV and Sine-Gordon breathers in the periodic and nonperiodic settings. Other rigorous stability results for breathers can be found in [6,7,18,16,3].…”

Section: Introductionmentioning

confidence: 99%

The Akhmediev breather is unstable

Alejo

Fanelli

Muñoz

2019

São Paulo J. Math. Sci.

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In this note, we give a rigorous proof that the NLS periodic Akhmediev breather is unstable. The proof follows the ideas in [17], in the sense that a suitable modification of the Stokes wave is the global attractor of the local Akhmediev dynamics for sufficiently large time, and therefore the latter cannot be stable in any suitable finite energy periodic Sobolev space.1991 Mathematics Subject Classification. 35Q55, 35Q51; 35Q35, 35Q40.

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Nonlinear Stability of MKdV Breathers

Cited by 56 publications

References 46 publications

Breathers and the Dynamics of Solutions in KdV Type Equations

Breathers and the Dynamics of Solutions in KdV Type Equations

On the Dynamics of Zero-Speed Solutions for Camassa–Holm-Type Equations

The Akhmediev breather is unstable

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