Discretizations, integrable systems and computation
Discretizations and associated numerical computation of solutions of certain integrable systems, such as the nonlinear Schrödinger equation (NLS) and the sine-Gordon (sG) equations with periodic boundary values can lead to instabilities, chaotic and spurious results. The chaos can be due to truncation errors or even roundoff errors and can be traced to the fact that these integrable systems are strongly unstable when the initial values are in the neighbourhood of homoclinic manifolds. By using the associated nonlinear spectral transform of the NLS equation and tracking the evolution of relevant eigenvalues one can observe and relate crossing of homoclinic manifolds to the temporal chaos in the waveforms when the initial data is even. For general initial values, even though there is no crossing of the unperturbed homoclinic manifolds, the waveforms still exhibit chaotic phenomena which can be related to the evolution of the spectrum. This paper reviews the current understanding of this intriguing phenomena and also compares the implementation of certain symplectic integrators and Runge-Kutta algorithms for the NLS and sG equations in regions of phase space proximate to the homoclinic manifolds.