On uniqueness properties of solutions of the Toda and Kac-van Moerbeke hierarchies

TL;DR

This paper proves that solutions to the Toda and Kac-van Moerbeke hierarchies cannot decay too rapidly at two different times unless they are trivial, highlighting a uniqueness property of these integrable systems.

Contribution

It establishes a new uniqueness theorem for solutions of the Toda and Kac-van Moerbeke hierarchies based on decay properties at different times.

Findings

Solutions with rapid decay at two times are trivial.

The result applies to entire hierarchies, not just single equations.

Provides a new perspective on the decay behavior of integrable systems.

Abstract

We prove that a solution of the Toda lattice cannot decay too fast at two different times unless it is trivial. In fact, we establish this result for the entire Toda and Kac-van Moerbeke hierarchies.