### Abstract

The inverse spectral transform for integrable nonlinear ordinary and partial differential equations (such as the Toda lattice, Korteweg-de Vries, sine-Gordon and nonlinear Schrödinger equations) provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions. These oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of N phases, ?_{j}(x, t) = ?_{j}x + ?_{j}t + ?^{0}_{j}, j = 1, ..., N, where the wave is 2p-periodic independently in each phase. This paper exhibits the numerical implementation of the inverse spectral solution of the sine-Gordon equation. The general construction is outlined and then implemented for N = 1, 2, and 3. We compute: (1) the exact theta-function solutions, (2) the Floquet spectrum of x-periodic solutions, (3) the labelling of linearized instabilities of N-phase solutions in terms of spectral data, and (4) the linearized growth rate in each unstable mode. The associated surfaces q_{N}(x, t) are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system. © 1991.

Original language | English |
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Pages (from-to) | 169-231 |

Number of pages | 63 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 48 |

Issue number | 1 |

Publication status | Published - Feb 1991 |

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## Cite this

*Physica D: Nonlinear Phenomena*,

*48*(1), 169-231.