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Atomistic to continuum models for crystals

By E. McMillan

Abstract

The theory of nonlinear mass-spring chains has a history stretching back to the now famous numerical simulations of Fermi, Pasta and Ulam. The unexpected results of that experiment have led to many new fields of study. Despite this, the mathematics of the lattice equations have proved sufficiently rich to attract continued attention to the present day. This work is concerned with the motions of an infinite one dimensional lattice with nearest-neighbour interactions governed by a generic potential. The Hamiltonian of such a system may be written $H = \sum_{i=-\infty}^{\infty} \, \Bigl(\frac{1}{2}p_i^2 + V(q_{i+1}-q_i)\Bigr)$, in terms of the momenta $p_i$ and the displacements $q_i$ of the lattice sites. All sites are assumed to be of equal mass. Certain generic conditions are placed on the potential $V$. Of particular interest are the solitary wave solutions which are known to exist upon such lattices. The KdV equation has long been known to emerge in a formal manner from the lattice equations as a continuum limit. More recently, the lattice's localized nonlinear modes have been rigorously approximated by the KdV's well-studied soliton solution, in the lattice's long wavelength regime. To date, however, little is known about how, and to what extent, lattice solitary waves differ from KdV solitons. It is proved in this work that a solution (which we prove to be unique) to a particular linear ordinary differential equation provides a correction to the KdV approximation. This gives, in an explicit way, the lowest order effect of lattice discreteness upon lattice solitary waves. It is also shown how such discreteness effects are propagated along the lattice both in isolation (single soliton case), and in the presence of another soliton correction (the bisoliton case). In the latter case their interaction is studied and the impact of lattice discreteness upon lattice solitary wave interactions is observed. This is possible by virtue of the discovery of an evolution equation for discreteness effects on the lattice. This equation is proved to have appropriate unique solutions and is found to be strikingly similar to corresponding equations known in both the theories of shallow water waves and ion-acoustic waves

Topics: Partial differential equations, Dynamical systems and ergodic theory
Year: 2003
OAI identifier: oai:generic.eprints.org:51/core69

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Citations

  1. (1997). A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge Texts in Applied Mathematics.
  2. (1981). A quasicontinuum approximation for solitons in an atomic chain.
  3. (1974). A scheme for integrating the nonlinear equations of mathematical physics by the method of the Inverse Scattering Problem. doi
  4. (1999). Analyse Fonctionnelle: Theorie et Applications. Dunod,
  5. (2001). Analysis,
  6. (1974). Complete integrability and stochastization of discrete dynamical systems.
  7. (1976). Contribution of higher order terms in the reductive perturbation theory I : A case of weakly dispersive wave.
  8. (1977). Contribution of the second order terms to the nonlinear shallow water waves.
  9. (1967). Difference Methods for Initial-Value Problems. Interscience, doi
  10. (1989). Discrete lattice solitons: properties and stability. doi
  11. (1991). Discrete model for DNA-promoter dynamics. doi
  12. (1977). Dynamical processes of the dressed ion acoustic solitons.
  13. (1986). Dynamics of nonlinear mass-spring chains near the continuum limit. doi
  14. (1975). Elementary Solid State Physics: Principles and Applications.
  15. (1973). Exact N-soliton solution of a nonlinear lumped network equation. doi
  16. (1971). Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. doi
  17. (1972). Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear media.
  18. (1994). Existence theorem for solitary waves on lattices. doi
  19. (1999). Functional Analysis in Applied Mathematics and Engineering.
  20. (1968). Integrals of nonlinear equations of evolution and solitary waves.
  21. (1974). Integrals of the Toda lattice. doi
  22. (1965). Interaction of solitons on a collisionless plasma and the recurrence of initial states. doi
  23. (1999). Inverse Scattering for the Toda hierarchy. doi
  24. (2002). KdV and KAM. Available at www.poschel.de,
  25. (1989). Mathematical Methods of Classical Mechanics.
  26. (1967). Method for solving the Korteweg-de Vries equation. doi
  27. (1953). Methods of Theoretical Physics: Part I.
  28. (2002). Multiscale correction to solitary wave solutions on FPU lattices.
  29. (2000). Near-integrability of FPU chains.
  30. (1998). Nonlinear Waves and Weak Turbulence. doi
  31. (1994). Numerical Hamiltonian Problems. doi
  32. (1974). On the Toda lattice.
  33. (1997). Postmodern Analysis.
  34. (1999). Quantum Chemistry. Prentice Hall, Upper Saddle River, N.J., fifth edition,
  35. (1991). Semi-discrete Fourier spectral approximations of infinite dimensional Hamiltonian systems and conservation laws. doi
  36. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations.
  37. (1990). Small divisors with spatial structure in infinite dimensional Hamiltonian systems.
  38. (2002). Solitary waves on FPU lattices II: Linear stability implies nonlinear stability. doi
  39. (1999). Solitary waves on FPU lattices: I: Qualitative properties, renormalization and continuum limit.
  40. (1997). Solitary waves with prescribed speed on infinite lattices. doi
  41. (1986). Soliton interactions (for the Korteweg-de Vries equation): a new perspective.
  42. (1997). Solitons: Nonlinear Pulses and Beams.
  43. (2000). Spectral Methods in MATLAB.
  44. (1982). Spectral Transform and Solitons: Tools to solve and investigate nonlinear evolution equations. Volume one.
  45. (1989). Stability and convergence at the PDE / stiff ODE interface. doi
  46. (1955). Studies of nonlinear problems. Los Alamos Sci.
  47. (2001). Symmetry and resonance in periodic FPU chains.
  48. (1974). The inverse scattering transform - Fourier analysis for nonlinear problems.
  49. (1984). The KAM theory of systems with short range interactions I. doi
  50. (1971). The Korteweg-de Vries equation as a completely integrable Hamiltonian system. doi
  51. (1971). The Korteweg-de Vries equation as a Hamiltonian system. doi
  52. (1960). The second approximation to cnoidal and solitary waves.
  53. (1972). The superperiod of the nonlinear weighted string (FPU) problem. doi
  54. (1981). Theory of Nonlinear Lattices. doi
  55. (1994). Theory of Solitons in Inhomogeneous Media.
  56. (1976). Toda lattice as an integrable system and the uniqueness of Toda’s potential. doi
  57. (2000). Travelling waves in a chain of coupled nonlinear oscillators. doi
  58. (2000). Travelling waves in the Fermi-Pasta-Ulam lattice.

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