The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities

Giannoulis, Johannes; Mielke, Alexander

doi:10.1088/0951-7715/17/2/011

Cited by 48 publications

(60 citation statements)

References 25 publications

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“…Moreover, in a steady motion where all thev n 's vanish, (3) specializes to (2). For these reasons the function V is referred to as the "equilibrium velocity function".…”

Section: Mathematical Modelmentioning

confidence: 99%

“…The rigorous proofs needed to show that the results of such formal calculations are meaningful (or not) are mathematically highly technical, and beyond the preparation of the typical engineering undergraduate student, e.g. see Giannoulis and Mielke [2]. This does not however mean that the student should therefore accept the model at face value, without some thought into whether the continuous model provides a reasonable representation of the underlying microscopic model.…”

Section: Introductionmentioning

confidence: 99%

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Macroscopic Limits of Microscopic Models

Abeyaratne

2014

International Journal of Mechanical Engineering Education

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The definitive, peer reviewed and edited version of this article is published in: [Rohan Abeyaratne], [2014] AbstractMany physical systems are comprised of several discrete elements, the equations of motion of each element being known. If the system has a large number of degrees of freedom, it may be possible to treat it as a continuous system. In this event, one might wish to derive the equations of motion of the continuous (macroscopic) system by taking a suitable limit of the equations governing the discrete (microscopic) system. The classical example of this involves a row of particles with each particle connected to its nearest neighbor by a linear spring, its continuum counterpart being a linearly elastic bar; see Figure 1.In a typical undergraduate engineering subject on, say Dynamics, the transition from a discrete system to a continuous system is usually carried out through a formal Taylor expansion of the terms of the discrete model about some reference configuration. The aim of this paper is to draw attention to the fact that a macroscopic model derived in this way should be examined critically in order to confirm that it provides a faithful representation of the underlying microscopic model. We use a specific (striking) example to make this point. In this example, a simple solution of the discrete model can be stable or unstable depending on the state of the system. However the corresponding solution of the continuous system is always unstable! We go on to show how the dispersion relations of the two models can be used to identify the source of the discrepancy and to suggest how one might modify the continuous model.

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“…Moreover, in a steady motion where all thev n 's vanish, (3) specializes to (2). For these reasons the function V is referred to as the "equilibrium velocity function".…”

Section: Mathematical Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Macroscopic Limits of Microscopic Models

Abeyaratne

2014

International Journal of Mechanical Engineering Education

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“…In particular, it is not obvious that the nlS equation (3.32) combined with the modulation ansatz (3.28) yields approximate solutions for the KG chain. However, the careful residual analysis from [GM04,GM06] provides rigorous justification results, and thus we can regard P 0 as an approximate invariant manifold.…”

Section: From Kg To Nlsmentioning

confidence: 99%

Lagrangian and Hamiltonian two-scale reduction

Giannoulis

Herrmann

Mielke

2008

Journal of Mathematical Physics

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Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system.In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions.In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.

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“…Importantly, as β → 0 the L ∞ norm of a wavepacket (2.10) remains constant, hence nonlinear effects in (1.1) remain strong. Evolution of wavepackets in problems which can be reduced to the form (1.1) were studied for a variety of equations in numerous physical and mathematical papers, mostly by asymptotic expansions with respect to a single small parameter similar to β, see [13], [16], [20], [22], [27], [33], [34], [41], [45], [47], [48] and references therein. We are interested in general properties of evolutionary systems of the form (1.1) with wavepacket initial data which hold for a wide class of nonlinearities and all values of the space dimensions d and the number 2J of the system components.…”

Section: Wavepackets and Their Basic Propertiesmentioning

confidence: 99%

Nonlinear Dynamics of a System of Particle-Like Wavepackets

Babin

Figotin

Instability in Models Connected With Fluid Flows I

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This work continues our studies of nonlinear evolution of a system of wavepackets. We study a wave propagation governed by a nonlinear system of hyperbolic PDE's with constant coefficients with the initial data being a multi-wavepacket. By definition a general wavepacket has a well defined principal wave vector, and, as we proved in previous works, the nonlinear dynamics preserves systems of wavepackets and their principal wave vectors. Here we study the nonlinear evolution of a special class of wavepackets, namely particle-like wavepackets. A particle-like wavepacket is of a dual nature: on one hand, it is a wave with a well defined principal wave vector, on the other hand, it is a particle in the sense that it can be assigned a well defined position in the space. We prove that under the nonlinear evolution a generic multi-particle wavepacket remains to be a multi-particle wavepacket with a high accuracy, and every constituting single particle-like wavepacket not only preserves its principal wave number but also it has a well-defined space position evolving with a constant velocity which is its group velocity. Remarkably the described properties hold though the involved single particle-like wavepackets undergo nonlinear interactions and multiple collisions in the space. We also prove that if principal wavevectors of multi-particle wavepacket are generic, the result of nonlinear interactions between different wavepackets is small and the approximate linear superposition principle holds uniformly with respect to the initial spatial positions of wavepackets.

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The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities

Cited by 48 publications

References 25 publications

Macroscopic Limits of Microscopic Models

Macroscopic Limits of Microscopic Models

Lagrangian and Hamiltonian two-scale reduction

Nonlinear Dynamics of a System of Particle-Like Wavepackets

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