Lagrangian and Hamiltonian two-scale reduction

Giannoulis, Johannes; Herrmann, Michael; Mielke, Alexander

doi:10.1063/1.2956487

Cited by 7 publications

(9 citation statements)

References 49 publications

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“…In other words, the scalar conservation law (2) which has Hamiltonian H(ȳ) =´R Φ(Dȳ) dξ and symplectic product ẏ, y ′ sympl = R y ′ Dẏ dξ. The analogies between the structures of lattice and PDE are -in view of the skew-symmetry of both ∇ and D -not surprising and exemplify the more general principles laid out in [10]. Moreover, if we replace D by the macroscopic difference operator…”

Section: A Lagrangian and Hamiltonian Structures For Scalar Conservatmentioning

confidence: 66%

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Oscillatory Waves in Discrete Scalar Conservation Laws

Herrmann

2012

Math. Models Methods Appl. Sci.

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We study Hamiltonian difference schemes for scalar conservation laws with monotone flux function and establish the existence of a three-parameter family of periodic travelling waves (wavetrains). The proof is based on an integral equation for the dual wave profile and employs constrained maximization as well as the invariance properties of a gradient flow. We also discuss the approximation of wavetrains and present some numerical results.

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Section: A Lagrangian and Hamiltonian Structures For Scalar Conservatmentioning

confidence: 66%

“…With (10) we have derived the dual formulation of ( 6); the main mathematical difference between both formulations will be discussed at the end of §3.1.…”

Section: Integral Equation For the Dual Profile And Normalizationmentioning

confidence: 99%

Oscillatory Waves in Discrete Scalar Conservation Laws

Herrmann

2012

Math. Models Methods Appl. Sci.

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“…Preservation of this Hamiltonian structure is not automatic, except if one uses specific derivation procedures [58,59]. The log-NLS equation is Hamiltonian, since it can be formally written…”

Section: Remark 22 the More Classical (And P-dependent) Nls Equationmentioning

confidence: 99%

Travelling breathers and solitary waves in strongly nonlinear lattices

James

2018

Phil. Trans. R. Soc. A.

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We study the existence of travelling breathers and solitary waves in the discrete -Schrödinger (DpS) equation. This model consists of a one-dimensional discrete nonlinear Schrödinger (NLS) equation with strongly nonlinear inter-site coupling (a discrete-Laplacian). The DpS equation describes the slow modulation in time of small amplitude oscillations in different types of nonlinear lattices, where linear oscillators are coupled to nearest-neighbours by strong nonlinearities. Such systems include granular chains made of discrete elements interacting through a Hertzian potential ( = 5/2 for contacting spheres), with additional local potentials or resonators inducing local oscillations. We formally derive three amplitude PDEs from the DpS equation when the exponent of nonlinearity is close to (and above) unity, i.e. for lying slightly above 2. Each model admits localized solutions approximating travelling breather solutions of the DpS equation. One model is the logarithmic NLS equation which admits Gaussian solutions, and the other is fully nonlinear degenerate NLS equations with compacton solutions. We compare these analytical approximations to travelling breather solutions computed numerically by an iterative method, and check the convergence of the approximations when [Formula: see text] An extensive numerical exploration of travelling breather profiles for = 5/2 suggests that these solutions are generally superposed on small amplitude non-vanishing oscillatory tails, except for particular parameter values where they become close to strictly localized solitary waves. In a vibro-impact limit where the parameter becomes large, we compute an analytical approximation of solitary wave solutions of the DpS equation.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

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“…We may interprete this as a kind of (approximate) invariant manifold, and the macroscopic equation describes the evolution on this manifold, the functions A defining kind of coordinates. We refer to [Mie91] for exact reductions of Hamiltonian systems and to [DHM06,GHM06a,Mie06b,GHM06b] for the full details.…”

Section: Alexander Mielke (Joint Work With Johannes Giannoulis and Michael Herrmann)mentioning

confidence: 99%

PDE and Materials

Ball¹,

James²,

Müller³

2007

Oberwolfach Rep.

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This meeting brought together a carefully selected group of mathematicians, physicists, material scientists and engineers to discuss and review the impact of modern mathematical methods on the understanding of material behaviour and the design of new materials. A key issue is that the material behaviour is determined by physical processes at many different spatial and temporal scales which cannot be simulataneously resolved by brute force computation. Central themes of the meeting were (spatial) microstructures, the relation between atomistic and continuum models, the treatment of multiple temporal scales, and the influence of randomness and the use of methods from statistical mechanics.

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Lagrangian and Hamiltonian two-scale reduction

Cited by 7 publications

References 49 publications

Oscillatory Waves in Discrete Scalar Conservation Laws

Oscillatory Waves in Discrete Scalar Conservation Laws

Travelling breathers and solitary waves in strongly nonlinear lattices

PDE and Materials

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