Combined breathing–kink modes in the FPU lattice

Wattis, Jonathan A. D.; Pickering, Andrew; Gordoa, Pilar R.

doi:10.1016/j.physd.2010.11.002

Cited by 4 publications

(3 citation statements)

References 19 publications

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“…The modified Volterra system is potentially much more interesting, since it supports breather solutions as well as travelling waves. In previous works we have also found connections between breathers in FPU and the mKdV system [9].…”

Section: Introductionsupporting

confidence: 51%

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Behaviour of the extended modified Volterra lattice–Reductions to generalised mKdV and NLS equations

Wattis

Gordoa

Pickering

2018

Communications in Nonlinear Science and Numerical Simulation

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We consider the first member of an extended modified Volterra lattice hierarchy. This system of equations is differential with respect to one independent variable and differential-delay with respect to a second independent variable. We use asymptotic analysis to consider the long wavelength limits of the system. By considering various magnitudes for the parameters involved, we derive reduced equations related to the modified Korteweg-de Vries and nonlinear Schrödinger equations. Highlights:• we analyse the behaviour of solutions of the extended modified Volterra lattice • we derive PDEs which are asymptotic approximations of the lattice • we find similarity solutions of these limiting PDEs • we show that in certain cases the PDEs can be transformed to mKdV or NLS

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Section: Introductionsupporting

confidence: 51%

“…We assume ǫ ≪ 1, and define 9) and postulate that the solution of (1.3) has the form q(x, t) = ǫe iθ F + ǫ 2 (G 0 + e iθ G 1 + e 2iθ G 2 ) + ǫ 3 (H 0 + H 1 e iθ + H 2 e 2iθ + H 3 e 3iθ . .…”

Section: Reduction Of the Modified Volterra Equation To Nlsmentioning

confidence: 99%

Behaviour of the extended modified Volterra lattice–Reductions to generalised mKdV and NLS equations

Wattis

Gordoa

Pickering

2018

Communications in Nonlinear Science and Numerical Simulation

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“…With the parity symmetry of the interaction potential V it can readily be seen that in our case c + = 0. Examples of interaction potentials which satisfy condition A are given by the β-FPUT system with W (x) = βx 4 /4, the periodic potential W (x) = 1 − cos x and the soft bounded potential W (x) = 1 − sech 2 x, [35], [36], [39], [40].…”

Section: Description Of the Systemmentioning

confidence: 99%

Existence of exponentially spatially localised breather solutions for lattices of nonlinearly coupled particles: Schauder's fixed point theorem approach

Hennig,

Karachalios

2021

Preprint

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The problem of showing the existence of localised modes in nonlinear lattices has attracted considerable efforts from the physical but also from the mathematical viewpoint where a rich variety of methods has been employed. In this paper we prove that a fixed point theory approach based on the celebrated Schauder's Fixed Point Theorem may provide a general method to establish concisely not only the existence of localised structures but also a required rate of spatial localisation. As a case study we consider lattices of coupled particles with nonlinear nearest neighbour interaction and prove the existence of exponentially spatially localised breathers under necessary non-resonant conditions accompanied with the proof of energy bounds of the solutions. The approach covers a physically significant class of potentials.

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