### Abstract

This is the second part of a two‐part series on forced lattice vibrations in which a semi‐infinite lattice of one‐dimensional particles {x_{n}}_{n≧1}, (Formula Presented.) is driven from one end by a particle x_{0}. This particle undergoes a given, periodically perturbed, uniform motion x0(t) = 2at + h(yt) where a and γ are constants and h(·) has period 2π. Results and notation from Part I are used freely and without further comment. Here the authors prove that sufficiently ample families of traveling‐wave solutions of the doubly infinite system (Formula Presented.) exist in the cases γ > γ_{1} and γ_{1} > γ > γ_{2} for general restoring forces F. In the case with Toda forces, F(x) = e^{x}, the authors prove that sufficiently ample families of traveling‐wave solutions exist for all k, γ_{k} > γ > γ_{k+1}. By a general result proved in Part I, this implies that there exist time‐periodic solutions of the driven system (i) with k‐phase wave asymptotics in n of the type (Formula Presented.) with k = 0 or 1 for general F and k arbitrary for F(x) = e^{x} (when k = 0, take γ_{0} = ∞ and X_{0} ≡ 0).

Original language | English (US) |
---|---|

Pages (from-to) | 1251-1298 |

Number of pages | 48 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 48 |

Issue number | 11 |

DOIs | |

State | Published - 1995 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*48*(11), 1251-1298. https://doi.org/10.1002/cpa.3160481103

**Forced lattice vibrations : Part II.** / Deift, Percy; Kriecherbauer, Thomas; Venakides, Stephanos.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 48, no. 11, pp. 1251-1298. https://doi.org/10.1002/cpa.3160481103

}

TY - JOUR

T1 - Forced lattice vibrations

T2 - Part II

AU - Deift, Percy

AU - Kriecherbauer, Thomas

AU - Venakides, Stephanos

PY - 1995

Y1 - 1995

N2 - This is the second part of a two‐part series on forced lattice vibrations in which a semi‐infinite lattice of one‐dimensional particles {xn}n≧1, (Formula Presented.) is driven from one end by a particle x0. This particle undergoes a given, periodically perturbed, uniform motion x0(t) = 2at + h(yt) where a and γ are constants and h(·) has period 2π. Results and notation from Part I are used freely and without further comment. Here the authors prove that sufficiently ample families of traveling‐wave solutions of the doubly infinite system (Formula Presented.) exist in the cases γ > γ1 and γ1 > γ > γ2 for general restoring forces F. In the case with Toda forces, F(x) = ex, the authors prove that sufficiently ample families of traveling‐wave solutions exist for all k, γk > γ > γk+1. By a general result proved in Part I, this implies that there exist time‐periodic solutions of the driven system (i) with k‐phase wave asymptotics in n of the type (Formula Presented.) with k = 0 or 1 for general F and k arbitrary for F(x) = ex (when k = 0, take γ0 = ∞ and X0 ≡ 0).

AB - This is the second part of a two‐part series on forced lattice vibrations in which a semi‐infinite lattice of one‐dimensional particles {xn}n≧1, (Formula Presented.) is driven from one end by a particle x0. This particle undergoes a given, periodically perturbed, uniform motion x0(t) = 2at + h(yt) where a and γ are constants and h(·) has period 2π. Results and notation from Part I are used freely and without further comment. Here the authors prove that sufficiently ample families of traveling‐wave solutions of the doubly infinite system (Formula Presented.) exist in the cases γ > γ1 and γ1 > γ > γ2 for general restoring forces F. In the case with Toda forces, F(x) = ex, the authors prove that sufficiently ample families of traveling‐wave solutions exist for all k, γk > γ > γk+1. By a general result proved in Part I, this implies that there exist time‐periodic solutions of the driven system (i) with k‐phase wave asymptotics in n of the type (Formula Presented.) with k = 0 or 1 for general F and k arbitrary for F(x) = ex (when k = 0, take γ0 = ∞ and X0 ≡ 0).

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UR - http://www.scopus.com/inward/citedby.url?scp=84990634413&partnerID=8YFLogxK

U2 - 10.1002/cpa.3160481103

DO - 10.1002/cpa.3160481103

M3 - Article

VL - 48

SP - 1251

EP - 1298

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 11

ER -