Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves

Chirilus‐Bruckner, Martina; Chong, Christopher; Schneider, Guido; Uecker, Hannes

doi:10.1016/j.jmaa.2008.05.101

Cited by 10 publications

(2 citation statements)

References 12 publications

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“…While the previous ansatz contains no internal microsctructure, in [SUW09] the interaction of two (weakly) amplitude modulated pulses of different group velocities c 1 = c 2 and time-independent amplitudes is considered in the chain (1.3) under the dispersive scalings τ = ε 2 t y = ε(γ − c 1,2 t), and it is proved that after interaction the amplitudes retain their shape but experience a shift of order O(ε) in position; see also [BF06]. Analogous results are obtained in [CBea07,CBea08] on a continuous one-dimensional string.…”

Section: Introductionmentioning

confidence: 60%

Interaction of modulated pulses in scalar multidimensional nonlinear lattices

Giannoulis

2010

Applicable Analysis

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We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitrary-range, nonlinear interaction potentials and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high order of the scaling parameter, thereby deducing the resonance and non-resonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets of up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.

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Section: Introductionmentioning

confidence: 60%

Interaction of modulated pulses in scalar multidimensional nonlinear lattices

Giannoulis

2010

Applicable Analysis

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“…Using the BT, one can construct, for instance, a 2-breather describing the elastic collision of two individual breathers. Such an interaction for small amplitude wave packets in the general KG setting (21) has striking similarities to the breather interaction: to leading order, the shape of the wave packets remains unchanged after collision, the only interaction effects being a shift of envelope and carrier waves (see [40] for KG with constant coefficients and [41] for KG with spatially periodic coefficients). In the case of large amplitudes, however, the situation is far more complex, as is attested also by the complexity of interactions of simpler structures such as kinks in non-integrable KG models [13][14][15][16][17][18].…”

Section: A Continuum Limit: Genuine Breathers and Modulating Pulsesmentioning

confidence: 97%

sine-Gordon Equation: From Discrete to Continuum

Chirilus-Bruckner,

Chong,

Kevrekidis

et al. 2014

Preprint

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In the present Chapter, we consider two prototypical Klein-Gordon models: the integrable sine-Gordon equation and the non-integrable φ 4 model. We focus, in particular, on two of their prototypical solutions, namely the kink-like heteroclinic connections and the time-periodic, exponentially localized in space breather waveforms. Two limits of the discrete variants of these models are contrasted: on the one side, the analytically tractable original continuum limit, and on the opposite end, the highly discrete, so-called anti-continuum limit of vanishing coupling. Numerical computations are used to bridge these two limits, as regards the existence, stability and dynamical properties of the waves. Finally, a recent variant of this theme is presented in the form of PT -symmetric Klein-Gordon field theories and a number of relevant results are touched upon.

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Nonlinear dynamics of modulated waves on graphene like quantum graphs

Gilg

Schneider

Uecker

2022

Mathematische Nachrichten

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We consider cubic Klein-Gordon equations on infinite two-dimensional periodic metric graphs having for instance the form of graphene. At non-Dirac points of the spectrum, with a multiple scaling expansion Nonlinear Schrödinger (NLS) equations can be derived in order to describe slow modulations in time and space of traveling wave packets. Here we justify this reduction by proving error estimates between solutions of the cubic Klein-Gordon equations and the associated NLS approximations. Moreover, we discuss the validity of the modulation equations appearing by the same procedure at the Dirac points of the spectrum.

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Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves

Cited by 10 publications

References 12 publications

Interaction of modulated pulses in scalar multidimensional nonlinear lattices

Interaction of modulated pulses in scalar multidimensional nonlinear lattices

sine-Gordon Equation: From Discrete to Continuum

Nonlinear dynamics of modulated waves on graphene like quantum graphs

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