Multi-travelling waves for the nonlinear Klein-Gordon equation

Côte, Raphaël; Martel, Yvan

doi:10.1090/tran/7303

Cited by 33 publications

(15 citation statements)

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“…Let us mention that, dealing with complex valued solutions of (NLKG), thus opening the possibility of considering stable solitons, Bellazzani, Ghimenti and Le Coz [1] obtained a similar existence result for (NLKG) in this particular stable case. We also notice that the previous theorem has been extended to solutions describing multi-bound states by Côte and Martel [7], that is to multi-traveling waves made of any number of decoupled general (excited) bound states. In the present paper, we will however only focus on (real valued) multi-solitary waves in the above sense.…”

Section: Theorem 11 ([9]mentioning

confidence: 91%

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On existence and uniqueness of asymptotic $N$-soliton-like solutions of the nonlinear klein-gordon equation

Friederich¹

2021

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A. We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in R 1+ , ≥ 1, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schrödinger equations, we obtain an -parameter family of solutions of (NLKG) which converges exponentially fast to a sum of given (unstable) solitons. For = 1, this family completely describes the set of solutions converging to the soliton considered; for ≥ 2, we prove uniqueness in a class with explicit algebraic rate of convergence.

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Section: Theorem 11 ([9]mentioning

confidence: 91%

“…Note that for a general nonlinearity and for ≥ 2, the operator possibly counts several and multiple negative eigenvalues. We refer to Côte and Martel [7] for the detail of the spectral properties in this case.…”

Section: Elements Of Spectral Theory Concerning (Nlkg) For Allmentioning

confidence: 99%

On existence and uniqueness of asymptotic $N$-soliton-like solutions of the nonlinear klein-gordon equation

Friederich¹

2021

Preprint

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“…A multi-breather associated to the sum given in (15) of solitons and breathers is a solution ∈ ([ * , +∞), 2 (R)), for a constant * > 0, of (mKdV) such that (16) lim…”

Section: Definitionmentioning

confidence: 99%

“…Theorem 2. Given solitons and breathers (9), (10) whose velocities (11) and ( 12) satisfy (13), there exists a multi-breather associated to given in (15). Moreover, ∈ ∞ (R × R) ∩ ∞ (R, (R)) for any 0 and there exists > 0 such that for any 0, there exists 1 and * > 0 such that,…”

Section: Definitionmentioning

confidence: 99%

On the uniqueness of multi-breathers of the modified Korteweg-de Vries equation

Semenov¹

2021

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We consider the modified Korteweg-de Vries equation (mKdV) and prove that given any sum of solitons and breathers of (mKdV) (with distinct velocities), there exists a solution of (mKdV) such that ( ) − ( ) → 0 when → +∞, which we call multi-breather. In order to do this, we work at the 2 level (even if usually solitons are considered at the 1 level). We will show that this convergence takes place in any space and that this convergence is exponentially fast in time. We also show that the constructed multi-breather is unique in two cases: in the class of solutions which converge to the profile faster than the inverse of a polynomial of a large enough degree in time (we will call this a super polynomial convergence), or (without hypothesis on the convergence rate), when all the velocities are positive.

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“…These equations are all nonlinear (focusing) dispersive equations, as are the nonlinear Schrödinger equation (NLS) [41,52,53] or the nonlinear Klein-Gordon equation (KG) [16,18], and share a common property: they all admit special solutions called solitons, a bump that translates with a constant velocity without deformation. However, (mKdV) enjoys a specific feature: it admits another class of special solutions called breathers, which we will describe below.…”

mentioning

confidence: 99%

Orbital stability of a sum of solitons and breathers of the modified Korteweg-De Vries equation

Semenov¹

2021

Preprint

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In this article, we prove that a sum of solitons and breathers of the modified Korteweg-de Vries equation (mKdV) is orbitally stable. The orbital stability is shown in 2 . More precisely, we will show that if a solution of (mKdV) is close enough to a sum of solitons and breathers with distinct velocities at = 0 in the 2 sense, then it stays close to this sum of solitons and breathers, up to space translations for solitons and space or phase translations for breathers.From this, we deduce the orbital stability of a multi-breather of (mKdV), constructed in [49]. As an application of the orbital stability and the formula for multi-breathers obtained by inverse scattering method [51], we deduce a result about uniqueness of multi-breathers.

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Multi-travelling waves for the nonlinear Klein-Gordon equation

Cited by 33 publications

References 50 publications

On existence and uniqueness of asymptotic $N$-soliton-like solutions of the nonlinear klein-gordon equation

On existence and uniqueness of asymptotic $N$-soliton-like solutions of the nonlinear klein-gordon equation

On the uniqueness of multi-breathers of the modified Korteweg-de Vries equation

Orbital stability of a sum of solitons and breathers of the modified Korteweg-De Vries equation

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