NUMERICAL INVERSE SPECTRAL TRANSFORM FOR THE PERIODIC SINE-GORDON EQUATION - THETA-FUNCTION SOLUTIONS AND THEIR LINEARIZED STABILITY

被引:28
作者
FLESCH, R [1 ]
FOREST, MG [1 ]
SINHA, A [1 ]
机构
[1] OHIO STATE UNIV,DEPT MATH,COLUMBUS,OH 43210
来源
PHYSICA D | 1991年 / 48卷 / 01期
基金
美国国家科学基金会;
关键词
D O I
10.1016/0167-2789(91)90058-H
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The inverse spectral transform for integrable nonlinear ordinary and partial differential equations (such as the Toda lattice, Korteweg-de Vries, sine-Gordon and nonlinear Schrodinger 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, theta-j(x,t) = kappa-(j)x + omega-(j)t + theta-j0, j = 1,..., N, where the wave is 2-pi-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 outline 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.
引用
收藏
页码:169 / 231
页数:63
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