Talk Abstracts

The one-dimensional cubic nonlinear Schrodinger (NLS) equation is a prototypical integrable system, characterized by the existence of a Lax pair. Motivated by integrable examples such as the Manakov system, in this talk, we study the integrability of two-component cubic NLS systems in one space dimension. By examining the existence of a fourth conserved quantity following mass, momentum, and energy, we find that general systems are classified into three cases. If a nontrivial fourth conserved quantity exists and the conserved energy has a non-degenerate quadratic part, the system is integrable and reducible to one of fifteen standard forms. If a fourth conserved quantity exists but the energy is degenerate, the system contains a closed single-component cubic NLS equation. Otherwise, no nontrivial fourth conserved quantity exists, indicating a failure of integrability.