Pöppe triple systems and integrable equations

Abstract

We construct the combinatorial Pöppe triple system, or ternary algebra, that underlies the non-commutative nonlinear Schrödinger (NLS) and modified Korteweg–de Vries (mKdV) hierarchy. We demonstrate that the Pöppe triple system provides an effective and systematic procedure for establishing that the NLS and mKdV equations are directly linearisable, which, in principle, extends to the whole hierarchy. This naturally extends the combinatorial Pöppe algebra, recently used to constructively establish integrability and uniqueness for the whole non-commutative potential Korteweg–de Vries hierarchy, to the NLS and mKdV hierarchy.

Keywords

• Nonlinear Schrödinger equation
• Modified Korteweg–de Vries equation
• Triple system