Using the finite domain remnant of the continuous spectrum to examine integrability: Effect of boundary conditions

Abstract

The aim of this work is to propose a method for testing the integrability of a model partial differential (PDE) and/or differential difference equation (DDE), by examining it in a finite but large domain. For monoparametric families of PDE/DDE’s, that are known to possess isolated integrable points, we find that very special features occur in the finite domain remnant of the continuous (“phonon”) spectrum at these “singular” points. We identify these features in the case example of a PDE and a DDE (that sustain front and pulselike solutions, respectively) for different types of boundary conditions. The key finding of the work is that such spectral features are generic near the singular, integrable points and hence we propose to explore a given PDE/DDE in a finite but large domain for such traits, as a means of assessing its potential integrability.