Integrable-Hierarchy Derivative-Order Geometry: Multitime Tau Ladders, KP/Toda Hirota Closures, and Finite Plücker Recovery

Key Points

• The aim is to enhance derivative-order geometry by connecting it with integrable hierarchies through multitime tau ladders.
• Introduced multitime tau ladders to track derivative orders along multiple times.
• Established a universal cube law for the representation of hierarchies.
• Derived explicit KP/Toda closures using Hirota's approaches.
• Formulated a finite Plücker recovery theorem to demonstrate jet recovery of the derivative-order spectrum.
• Successfully linked derivative-order geometry to integrable hierarchies through the proposed methods.
• Showed that Hirota bilinear transmutation can generate the full hierarchy of derivatives.
• Demonstrated that a finite jet is sufficient to recover the entire derivative-order spectrum.