Global Existence in Infinite Lattices of Nonlinear Oscillators: The Discrete Klein-Gordon Equation

Karachalios, Nikos I.

doi:10.1017/s001708950600317x

Cited by 5 publications

(10 citation statements)

References 35 publications

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“…If µ is a Floquet multiplier, then the corresponding Floquet exponent λ (which is unique modulo 2πi/T ) is related to µ by µ = e λT . For every Floquet exponent λ, there is a corresponding solution v(t) = e λt w(t) to the linearized equation (7), where w(t) is periodic with period T (see, for example, [50,Lemma 2.1.29]). Substituting this ansatz into (7), we obtain the Floquet eigenvalue problem…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…For every Floquet exponent λ, there is a corresponding solution v(t) = e λt w(t) to the linearized equation (7), where w(t) is periodic with period T (see, for example, [50,Lemma 2.1.29]). Substituting this ansatz into (7), we obtain the Floquet eigenvalue problem…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…In this work, we will examine the discrete Klein-Gordon (DKG) equation [3][4][5][6] ün = d(∆ 2 u) n − f (u n ), which describes the dynamics of an infinitely long, one-dimensional lattice of particles. The quantity u n represents the displacement of nth particle in the integer lattice as a function of the time t. Each particle is harmonically coupled to its two nearest neighbors via the discrete second difference operator ∆ 2 , and the strength of this coupling is quantified by the parameter d. The particles are subject to an external, nonlinear, on-site potential V (u), such that f (u) = V ′ (u), which can be of different type depending on the model [7,8]. Common nonlinearities are shown in Table 1.…”

Section: Introductionmentioning

confidence: 99%

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Revisiting Multi-breathers in the discrete Klein-Gordon equation: A Spatial Dynamics Approach

Parker¹,

Cuevas–Maraver²,

Kevrekidis³

et al. 2022

Preprint

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We consider the existence and spectral stability of multi-breather structures in the discrete Klein-Gordon equation, both for soft and hard symmetric potentials. By taking a spatial dynamics approach and an appropriate approximation of the system on a finite-dimensional Hilbert space, we use Lin's method to construct multi-breathers from a sequence of well-separated copies of a primary, single-site breather. We then locate the eigenmodes in the Floquet spectrum associated with the interaction between the individual breathers of such multi-breather states by reducing the spectral problem to a matrix equation. Analytical expressions for these eigenmodes for the approximate system are obtained in terms of the primary breather and its kernel eigenfunctions, and these are found to be in very good agreement with the numerical Floquet spectrum results. This is supplemented with results from numerical timestepping experiments, which are interpreted using the spectral computations.

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Section: Mathematical Backgroundmentioning

confidence: 99%

Section: Mathematical Backgroundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Revisiting Multi-breathers in the discrete Klein-Gordon equation: A Spatial Dynamics Approach

Parker¹,

Cuevas–Maraver²,

Kevrekidis³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…which describes the dynamics of an infinitely long, one-dimensional lattice of particles. The quantity u n represents the displacement of the nth particle in the integer lattice as a function of the time t. Each particle is harmonically coupled to its two nearest neighbors via the discrete second difference operator Δ 2 , and the strength of this coupling is quantified by the parameter d. The particles are subject to an external, nonlinear, on-site potential V(u), such that f (u) = V (u), which can be of different type depending on the model [7,8]. Common nonlinearities are shown in table 1.…”

Section: Introductionmentioning

confidence: 99%

Revisiting multi-breathers in the discrete Klein–Gordon equation: a spatial dynamics approach

et al. 2022

View full text Add to dashboard Cite

We consider the existence and spectral stability of multi-breather structures in the discrete Klein–Gordon equation, both for soft and hard symmetric potentials. To obtain analytical results, we project the system onto a finite-dimensional Hilbert space consisting of the first M Fourier modes, for arbitrary M. On this approximate system, we then take a spatial dynamics approach and use Lin’s method to construct multi-breathers from a sequence of well-separated copies of the primary, single-site breather. We then locate the eigenmodes in the Floquet spectrum associated with the interaction between the individual breathers of such multi-breather states by reducing the spectral problem to a matrix equation. Expressions for these eigenmodes for the approximate, finite-dimensional system are obtained in terms of the primary breather and its kernel eigenfunctions, and these are found to be in very good agreement with the numerical Floquet spectrum results. This is supplemented with results from numerical timestepping experiments, which are interpreted using the spectral computations.

show abstract

“…describes the dynamics of an infinitely long, one-dimensional lattice of particles which are harmonically coupled to their neighbors through the discrete second difference operator ∆ 2 and are subject to an external, nonlinear, on-site potential P (u) such that f (u) = P ′ (u) [8].…”

Section: Introductionmentioning

confidence: 99%

Stationary multi-kinks in the discrete sine-Gordon equation

Parker,

Kevrekidis,

Aceves

2021

Preprint

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We consider the existence and spectral stability of static multi-kink structures in the discrete sine-Gordon equation, as a representative example of the family of discrete Klein-Gordon models. The multi-kinks are constructed using Lin's method from an alternating sequence of well-separated kink and antikink solutions. We then locate the point spectrum associated with these multi-kink solutions by reducing the spectral problem to a matrix equation. For an m-structure multi-kink, there will be m eigenvalues in the point spectrum near each eigenvalue of the primary kink, and, as long as the spectrum of the primary kink is imaginary, the spectrum of the multi-kink will be as well. We obtain analytic expressions for the eigenvalues of a multi-kink in terms of the eigenvalues and corresponding eigenfunctions of the primary kink, and these are in very good agreement with numerical results. We also perform numerical time-stepping experiments on perturbations of multikinks, and the outcomes of these simulations are interpreted using the spectral results.[44] Alouf Jirari, Second-order Sturm-Liouville difference equations and orthogonal polynomials, American Mathematical Society, 1995.

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Global Existence in Infinite Lattices of Nonlinear Oscillators: The Discrete Klein-Gordon Equation

Cited by 5 publications

References 35 publications

Revisiting Multi-breathers in the discrete Klein-Gordon equation: A Spatial Dynamics Approach

Revisiting Multi-breathers in the discrete Klein-Gordon equation: A Spatial Dynamics Approach

Revisiting multi-breathers in the discrete Klein–Gordon equation: a spatial dynamics approach

Stationary multi-kinks in the discrete sine-Gordon equation

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