### Abstract

The purpose of this paper is the derivation of reduced, finite-dimensional dynamical systems that govern the near-integrable modulations of N-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to any N-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation order N. We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study.

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

Pages (from-to) | 393-426 |

Number of pages | 34 |

Journal | Journal of Nonlinear Science |

Volume | 3 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1993 |

### Fingerprint

### Keywords

- modulation equations
- nearly integrable PDE
- nonlinear modes
- numerical simulation

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Applied Mathematics
- Mathematics(all)
- Mechanics of Materials
- Computational Mechanics

### Cite this

*Journal of Nonlinear Science*,

*3*(1), 393-426. https://doi.org/10.1007/BF02429871

**Strongly nonlinear modal equations for nearly integrable PDEs.** / Ercolani, Nicholas M; Forest, M. G.; McLaughlin, D. W.; Sinha, A.

Research output: Contribution to journal › Article

*Journal of Nonlinear Science*, vol. 3, no. 1, pp. 393-426. https://doi.org/10.1007/BF02429871

}

TY - JOUR

T1 - Strongly nonlinear modal equations for nearly integrable PDEs

AU - Ercolani, Nicholas M

AU - Forest, M. G.

AU - McLaughlin, D. W.

AU - Sinha, A.

PY - 1993/12

Y1 - 1993/12

N2 - The purpose of this paper is the derivation of reduced, finite-dimensional dynamical systems that govern the near-integrable modulations of N-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to any N-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation order N. We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study.

AB - The purpose of this paper is the derivation of reduced, finite-dimensional dynamical systems that govern the near-integrable modulations of N-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to any N-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation order N. We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study.

KW - modulation equations

KW - nearly integrable PDE

KW - nonlinear modes

KW - numerical simulation

UR - http://www.scopus.com/inward/record.url?scp=51249162979&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249162979&partnerID=8YFLogxK

U2 - 10.1007/BF02429871

DO - 10.1007/BF02429871

M3 - Article

VL - 3

SP - 393

EP - 426

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 1

ER -