TY - JOUR
T1 - Numerical inverse spectral transform for the periodic sine-gordon equation
T2 - Theta function solutions and their linearized stability
AU - Flesch, Randy
AU - Forest, M. Gregory
AU - Sinha, Amarendra
PY - 1991/2
Y1 - 1991/2
N2 - 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 + ?0j, 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 qN(x, t) are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system. © 1991.
AB - 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 + ?0j, 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 qN(x, t) are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system. © 1991.
UR - http://www.scopus.com/inward/record.url?scp=0011347758&partnerID=8YFLogxK
M3 - Article
SN - 0167-2789
VL - 48
SP - 169
EP - 231
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
IS - 1
ER -