Numerical inverse spectral transform for the periodic sine-gordon equation: Theta function solutions and their linearized stability

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) = ?_jx + ?_jt + ?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.