### Abstract

The phenomenon of spatio-temporal phase modulation made possible by resonance is investigated in detail through the analysis of an example problem. A simple family of exact solutions to the Ablowitz-Ladik equations is found to be modulationally stable in some regimes. This family of solutions is determined by fixing antiperiod 2 boundary conditions, which determines two wavenumbers. Within the family of solutions, the frequencies do not depend on amplitude; this feature ensures that the antiperiod 2 boundary conditions will be enforced under modulation. The family of solutions is described by four parameters, two being actions that foliate the phase space, and two being macroscopically observable functions of the phase constants. The modulation of the actions is described by a closed hyperbolic system of first order equations, which is consistent with the full set of four genus 1 modulation equations. The modulation of the phase information, easily observed due to the presence of two resonances, is described by two more equations that are driven by the actions. The results are confirmed by numerical experiments.

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

Pages (from-to) | 1-27 |

Number of pages | 27 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 92 |

Issue number | 1-2 |

State | Published - 1996 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*92*(1-2), 1-27.

**Modulation of multiphase waves in the presence of resonance.** / Miller, Peter D.; Ercolani, Nicholas M; Levermore, C. David.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 92, no. 1-2, pp. 1-27.

}

TY - JOUR

T1 - Modulation of multiphase waves in the presence of resonance

AU - Miller, Peter D.

AU - Ercolani, Nicholas M

AU - Levermore, C. David

PY - 1996

Y1 - 1996

N2 - The phenomenon of spatio-temporal phase modulation made possible by resonance is investigated in detail through the analysis of an example problem. A simple family of exact solutions to the Ablowitz-Ladik equations is found to be modulationally stable in some regimes. This family of solutions is determined by fixing antiperiod 2 boundary conditions, which determines two wavenumbers. Within the family of solutions, the frequencies do not depend on amplitude; this feature ensures that the antiperiod 2 boundary conditions will be enforced under modulation. The family of solutions is described by four parameters, two being actions that foliate the phase space, and two being macroscopically observable functions of the phase constants. The modulation of the actions is described by a closed hyperbolic system of first order equations, which is consistent with the full set of four genus 1 modulation equations. The modulation of the phase information, easily observed due to the presence of two resonances, is described by two more equations that are driven by the actions. The results are confirmed by numerical experiments.

AB - The phenomenon of spatio-temporal phase modulation made possible by resonance is investigated in detail through the analysis of an example problem. A simple family of exact solutions to the Ablowitz-Ladik equations is found to be modulationally stable in some regimes. This family of solutions is determined by fixing antiperiod 2 boundary conditions, which determines two wavenumbers. Within the family of solutions, the frequencies do not depend on amplitude; this feature ensures that the antiperiod 2 boundary conditions will be enforced under modulation. The family of solutions is described by four parameters, two being actions that foliate the phase space, and two being macroscopically observable functions of the phase constants. The modulation of the actions is described by a closed hyperbolic system of first order equations, which is consistent with the full set of four genus 1 modulation equations. The modulation of the phase information, easily observed due to the presence of two resonances, is described by two more equations that are driven by the actions. The results are confirmed by numerical experiments.

UR - http://www.scopus.com/inward/record.url?scp=22244445849&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=22244445849&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:22244445849

VL - 92

SP - 1

EP - 27

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -