Numerical chaos, roundoff errors, and homoclinic manifolds

Abstract

The focusing nonlinear Schrödinger equation is numerically integrated over moderate to long time intervals. In certain parameter regimes small errors on the order of roundoff grow rapidly and saturate at values comparable to the main wave. Although the constants of motion are nearly preserved, a serious phase instability (chaos) develops in the numerical solutions. The instability is found to be associated with homoclinic structures and the underlying mechanisms apply equally well to many Hamiltonian wave systems.