Uniqueness of solutions of the KdV-hierarchy via Dubrovin-type flows

TL;DR

This paper proves the uniqueness of solutions to the KdV hierarchy for various initial data types, including reflectionless, almost periodic, and small quasiperiodic, by analyzing Dubrovin-type flows and trace formulas.

Contribution

It establishes the Lipschitz continuity of Dirichlet data evolution and uniqueness of solutions for the KdV hierarchy under broad spectral conditions, extending previous results.

Findings

Reflectionless solutions remain reflectionless over time.

Dirichlet data evolve according to Lipschitz Dubrovin-type flows.

Uniqueness of solutions for periodic initial data without Craig-type conditions.

Abstract

We consider the Cauchy problem for the KdV hierarchy -- a family of integrable PDEs with a Lax pair representation involving one-dimensional Schr\"odinger operators -- under a local in time boundedness assumption on the solution. For reflectionless initial data, we prove that the solution stays reflectionless. For almost periodic initial data with absolutely continuous spectrum, we prove that under Craig-type conditions on the spectrum, Dirichlet data evolve according to a Lipschitz Dubrovin-type flow, so the solution is uniquely recovered by a trace formula. This applies to algebro-geometric (finite gap) solutions; more notably, we prove that it applies to small quasiperiodic initial data with analytic sampling functions and Diophantine frequency. This also gives a uniqueness result for the Cauchy problem on the line for periodic initial data, even in the absence of Craig-type conditions.