### Abstract

An analytical study is conducted of the structure, stability, and bifurcation of the spatially dependent time-periodic solutions of the damped-driven sine-Gordon equation in the nonlinear Schrodinger approximation. Locked states are found for which the spatial structure consists of coherent excitations localized about x = 0 or L/2. A bifurcation analysis reveals the relationship of these spatially localized solutions to the spatially independent ones and provides a cutoff wavenumber above which there are no spatially dependent solutions; this establishes an upper bound on the number of local excitations comprising the spatial pattern. A linear stability analysis shows that the spatially localized solutions undergo a Hopf bifurcation to temporal quasi-periodicity as the driver amplitude Γ is increased. For sufficiently high driver frequencies, the temporally periodic solution regains its stability (via another Hopf bifurcation) in a Γ-window of finite width before undergoing a third Hopf bifurcation to quasi-periodicity. The analytical results compare favorably with numerical solutions and provide the requisite ingredients for construction of chaotic attractors for this system.

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

Pages (from-to) | 791-818 |

Number of pages | 28 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 50 |

Issue number | 3 |

State | Published - Jun 1990 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Applied Mathematics*,

*50*(3), 791-818.

**Stability and bifurcation of spatially coherent solutions of the damped-driven NLS equation.** / Terrones, Guillermo; McLaughlin, David W.; Overman, Edward A.; Pearlstein, Arne J.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 50, no. 3, pp. 791-818.

}

TY - JOUR

T1 - Stability and bifurcation of spatially coherent solutions of the damped-driven NLS equation

AU - Terrones, Guillermo

AU - McLaughlin, David W.

AU - Overman, Edward A.

AU - Pearlstein, Arne J.

PY - 1990/6

Y1 - 1990/6

N2 - An analytical study is conducted of the structure, stability, and bifurcation of the spatially dependent time-periodic solutions of the damped-driven sine-Gordon equation in the nonlinear Schrodinger approximation. Locked states are found for which the spatial structure consists of coherent excitations localized about x = 0 or L/2. A bifurcation analysis reveals the relationship of these spatially localized solutions to the spatially independent ones and provides a cutoff wavenumber above which there are no spatially dependent solutions; this establishes an upper bound on the number of local excitations comprising the spatial pattern. A linear stability analysis shows that the spatially localized solutions undergo a Hopf bifurcation to temporal quasi-periodicity as the driver amplitude Γ is increased. For sufficiently high driver frequencies, the temporally periodic solution regains its stability (via another Hopf bifurcation) in a Γ-window of finite width before undergoing a third Hopf bifurcation to quasi-periodicity. The analytical results compare favorably with numerical solutions and provide the requisite ingredients for construction of chaotic attractors for this system.

AB - An analytical study is conducted of the structure, stability, and bifurcation of the spatially dependent time-periodic solutions of the damped-driven sine-Gordon equation in the nonlinear Schrodinger approximation. Locked states are found for which the spatial structure consists of coherent excitations localized about x = 0 or L/2. A bifurcation analysis reveals the relationship of these spatially localized solutions to the spatially independent ones and provides a cutoff wavenumber above which there are no spatially dependent solutions; this establishes an upper bound on the number of local excitations comprising the spatial pattern. A linear stability analysis shows that the spatially localized solutions undergo a Hopf bifurcation to temporal quasi-periodicity as the driver amplitude Γ is increased. For sufficiently high driver frequencies, the temporally periodic solution regains its stability (via another Hopf bifurcation) in a Γ-window of finite width before undergoing a third Hopf bifurcation to quasi-periodicity. The analytical results compare favorably with numerical solutions and provide the requisite ingredients for construction of chaotic attractors for this system.

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M3 - Article

AN - SCOPUS:0025448184

VL - 50

SP - 791

EP - 818

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 3

ER -