Breather Solutions in Periodic Media

Blank, Carsten; Chirilus‐Bruckner, Martina; Lescarret, Vincent; Schneider, Guido

doi:10.1007/s00220-011-1191-3

Cited by 26 publications

(67 citation statements)

References 23 publications

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“…We follow a similar strategy as stated in [8]. It is enough to prove that, under the orthogonality conditions (4.25), and the additional constraint 12 In other words, we control the variations of the scaling parameter β at least at the second order in the error term (z, w). and we can assume…”

Section: Spectral Analysismentioning

confidence: 99%

“…It is also relevant to mention that the construction of a breather solution with different patterns as the usually required can be very involved. For example, a different class of periodic breather was discovered by Blank et al in [12]. This particular solution is constructed for the so called φ 4 model, by assuming that the equation is no longer autonomous, but it has suitable, well-chosen periodic coefficients.…”

Section: 3mentioning

confidence: 99%

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On the variational structure of breather solutions I: Sine-Gordon equation

Alejo¹,

Muñoz²,

Palacios³

2017

Journal of Mathematical Analysis and Applications

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In this paper we give a systematic and simple account that put in evidence that many breather solutions of integrable equations satisfy suitable variational elliptic equations, which also implies that the stability problem reduces in some sense to (i) the study of the spectrum of explicit linear systems (spectral stability), and (ii) the understanding of how bad directions (if any) can be controlled using low regularity conservation laws. We exemplify this idea in the case of the modified Korteweg-de Vries (mKdV), Gardner, and the more involved sine-Gordon (SG) equation. Then we perform numerical simulations that confirm, at the level of the spectral problem, our previous rigorous results in [8,10], where we showed that mKdV breathers are H 2 and H 1 stable, respectively. In a second step, we also discuss the Gardner case, a relevant modification of the KdV and mKdV equations, recovering similar results. Then we discuss the Sine-Gordon case, where the spectral study of a fourth-order linear matrix system is the key element to show stability. Using numerical methods, we confirm that all spectral assumptions leading to the H 2 × H 1 stability of SG breathers are numerically satisfied, even in the ultra-relativistic, singular regime. In a second part, we study the periodic mKdV case, where a periodic breather is known from the work of Kevrekidis et al. [41]. We rigorously show that these breathers satisfy a suitable elliptic equation, and we also show numerical spectral stability. However, we also identify the source of nonlinear instability in the case described in [41], and we conjecture that, even if spectral stability is satisfied, nonlinear stability/instability depends only on the sign of a suitable discriminant function, a condition that is trivially satisfied in the case of non-periodic breathers. Finally, we present a new class of breather solution for mKdV, believed to exist from geometric considerations, and which is periodic in time and space, but has nonzero mean, unlike standard breathers.

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Section: Spectral Analysismentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

On the variational structure of breather solutions I: Sine-Gordon equation

Alejo¹,

Muñoz²,

Palacios³

2017

Journal of Mathematical Analysis and Applications

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“…Recently, Kowalczyk, Martel and the first author [20] showed nonexistence of odd breathers for scalar field equations with odd nonlinearities, with no other assumptions on the nonlinearity, except being C 1 . However, in [7] it was shown existence of breathers in scalar field equations with non-homogeneous coefficients. Finally, [31] considers in a rigorous way the stability question for Peregrine and Ma breathers, showing that they are indeed unstable, even if the equation is locally well-posed.…”

Section: -149])mentioning

confidence: 99%

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

Muñoz

Palacios

2019

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

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In this article we prove that 2-soliton solutions of the sine-Gordon equation (SG) are orbitally stable in the natural energy space of the problem. The solutions that we study are the 2-kink, kink-antikink and breather of SG. In order to prove this result, we will use Bäcklund transformations implemented by the Implicit Function Theorem. These transformations will allow us to reduce the stability of the three solutions to the case of the vacuum solution, in the spirit of previous results by Alejo and the first author [3], which was done for the case of the scalar modified Korteweg-de Vries equation. However, we will see that SG presents several difficulties because of its vector valued character. Our results improve those in [5], and give a first rigorous proof of the stability in the energy space of SG 2-solitons.

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“…With regard to the decay behavior in k and the ensuing integration by parts, we record Lemma 4. 5 We have the following estimates:…”

Section: Analysis Of the High-energy Regime Away From The Thresholdsmentioning

confidence: 99%

“… The knowledge about the stability properties of spatially localized structures in linear periodic media with and without defects is fundamental for many fields in nature. Its importance for the design of photonic crystals is, for example, described in and . Against this background, we consider a one‐dimensional linear Klein‐Gordon equation to which both a spatially periodic Lamé potential and a spatially localized perturbation are added.…”

mentioning

confidence: 99%

Dispersive estimates for solutions to the perturbed one‐dimensional Klein‐Gordon equation with and without a one‐gap periodic potential

Prill

2014

Mathematische Nachrichten

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The knowledge about the stability properties of spatially localized structures in linear periodic media with and without defects is fundamental for many fields in nature. Its importance for the design of photonic crystals is, for example, described in and . Against this background, we consider a one‐dimensional linear Klein‐Gordon equation to which both a spatially periodic Lamé potential and a spatially localized perturbation are added. Given the dispersive character of the underlying equation, it is the purpose of this paper to deduce time‐decay rates for its solutions. We show that, generically, the part of the solution which is orthogonal to possible eigenfunctions of the perturbed Hill operator associated to the problem decays with a rate of t−13 w.r.t. the L∞ norm. In weighted L2 norms, we even get a time decay of t−32. Furthermore, we consider the situation of a perturbing potential that is only made up of a spatially localized part which, now, can be slightly more general. It is well‐known that, in general, it is not possible to obtain the L∞ endpoint estimate in one space dimension by means of the wave operators drawn from scattering theory. For this reason, we proceed directly and prove, along the lines of , the expected decay rate of t−12.

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Breather Solutions in Periodic Media

Cited by 26 publications

References 23 publications

On the variational structure of breather solutions I: Sine-Gordon equation

On the variational structure of breather solutions I: Sine-Gordon equation

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

Dispersive estimates for solutions to the perturbed one‐dimensional Klein‐Gordon equation with and without a one‐gap periodic potential

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