Counter-propagating Waves on Fluid Surfaces and the Continuum Limit of the Fermi-Pasta-Ulam Model

Schneider, Guido; Wayne, C. Eugene

doi:10.1142/9789812792617_0075

Cited by 68 publications

(98 citation statements)

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“…We note that this improves on the estimate of k 3.5 which appears e.g. in [47,52]. [A number of details towards making this argument rigorous are presented in the Appendix].…”

Section: Connecting the Granular Chain And Its Soliton Collisionsupporting

confidence: 67%

“…It is known, both formally [46] and rigorously [47] (on long but finite time scales) that KdV approximates FPU α-type lattices for smallamplitude, long-wave, low-energy initial data. This fact has been used in the mathematical literature to determine the shape [48] and dynamical stability [49][50][51] of solitary waves and even of their interactions [52].…”

Section: Introductionmentioning

confidence: 99%

“…Since Hamiltonian lattices are timereversible, a single KdV equation cannot capture the evolution of general initial data. It is typical to use a pair of uncoupled KdV equations, one moving rightward and one moving leftward to capture the evolution of general initial data [47]. On the other hand, the Toda lattice has several benefits as an approximation of the granular problem.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Characterizing traveling-wave collisions in granular chains starting from integrable limits: The case of the Korteweg–de Vries equation and the Toda lattice

et al. 2014

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Our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression. To that effect, we aim to quantify approximations of the relevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV) equation and the Toda lattice. Using the availability in such settings of both 1-soliton and 2-soliton solutions in explicit analytical form, we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable (KdV or Toda) model. While the KdV offers the possibility to accurately capture collisions of solitary waves propagating in the same direction, the Toda lattice enables capturing both copropagating and counter-propagating soliton collisions. The error in the approximation is quantified numerically and connections to bounds established in the mathematical literature are also given.

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“…We note that this improves on the estimate of k 3.5 which appears e.g. in [47,52]. [A number of details towards making this argument rigorous are presented in the Appendix].…”

Section: Connecting the Granular Chain And Its Soliton Collisionsupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Characterizing traveling-wave collisions in granular chains starting from integrable limits: The case of the Korteweg–de Vries equation and the Toda lattice

et al. 2014

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“…An open question is to check that well-prepared initial data evolve (up to higher order terms and on long finite times) according to the log-KdV or H-KdV equation (in the same spirit as the justification of the classical KdV equation for FPU chains [15][16][17]). This problem may be extended to Hertzian FPU system (1.1) and (1.6), at least close to a solitary wave solution and for a suitable topology (i.e.…”

Section: Discussionmentioning

confidence: 99%

Gaussian solitary waves and compactons in Fermi–Pasta–Ulam lattices with Hertzian potentials

James

Pelinovsky

2014

Proc. R. Soc. A.

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We consider a class of fully nonlinear FermiPasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order α > 1. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyse the propagation of localized waves when α is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic Korteweg-de Vries (KdV) equation and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with Hölder-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When α → 1 + , we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile.

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“…Finally, we mention that rigorous justification results for the KdV reduction can be found in [SW00,FP99]. However, these results do not use the reduced Lagrangian or Hamiltonian structures, but work on the equation of motion directly.…”

Section: From Fpu To Kdvmentioning

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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Counter-propagating Waves on Fluid Surfaces and the Continuum Limit of the Fermi-Pasta-Ulam Model

Cited by 68 publications

References 0 publications

Characterizing traveling-wave collisions in granular chains starting from integrable limits: The case of the Korteweg–de Vries equation and the Toda lattice

Characterizing traveling-wave collisions in granular chains starting from integrable limits: The case of the Korteweg–de Vries equation and the Toda lattice

Gaussian solitary waves and compactons in Fermi–Pasta–Ulam lattices with Hertzian potentials

Lagrangian and Hamiltonian two-scale reduction

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